S´eminaire Bourbaki Juin 2000 52`emeann´ee, 1999-2000, no 875

VERTEX ALGEBRAS AND ALGEBRAIC CURVES

by EDWARD FRENKEL

1. INTRODUCTION

Vertex operators appeared in the early days of string theory as local operators describing propagation of string states. Mathematical analogues of these operators were discovered in representation theory of affine Kac-Moody algebras in the works of Lepowsky–Wilson [LW] and I. Frenkel–Kac [FK]. In order to formalize the emerging structure and moti- vated in particular by the I. Frenkel–Lepowsky–Meurman construction of the Moonshine Module of the Monster group, Borcherds gave the definition of vertex algebra in [B1]. The foundations of the theory were subsequently laid down in [FLM, FHL]; in particular, it was shown in [FLM] that the Moonshine Module indeed possessed a vertex algebra structure. In the meantime, Belavin, Polyakov and Zamolodchikov [BPZ] initiated the study of two-dimensional conformal field theory (CFT). Vertex algebras can be seen in retrospect as the mathematical equivalent of the central objects of CFT called the chi- ral symmetry algebras. Moreover, the key property of associativity of vertex algebras is equivalent to the property of operator product expansion in CFT, which goes back to the pioneering works of Polyakov and Wilson. Thus, vertex algebras may be thought of as the mathematical language of two-dimensional conformal field theory. Vertex algebras have turned out to be extremely useful in many areas of mathematics. They are by now ubiquitous in representation theory of infinite-dimensional Lie algebras. They have also found applications in such fields as algebraic geometry, theory of finite groups, modular functions and topology. Recently Beilinson and Drinfeld have introduced a remarkable geometric version of vertex algebras which they called chiral algebras [BD3]. Chiral algebras give rise to some novel concepts and techniques which are likely to have a profound impact on algebraic geometry. In this talk we review the theory of vertex algebras with a particular emphasis on their algebro-geometric interpretation and applications. We start in §2 with the axiomatic definition of vertex algebra, which is somewhat different from, but equivalent to Borcherds’ 875-02 original definition (see [DL, FKRW, K2]). We then discuss some of their most important properties and give the first examples, which come from infinite-dimensional Lie algebras, such as Heisenberg, affine Kac-Moody and the Virasoro algebra. More unconventional examples of vertex algebras which are not generated by a Lie algebra, such as the W– algebras, are reviewed in §3. In §4 we explain how to make vertex algebras coordinate independent, thus effectively getting rid of the formal variable omnipresent in the standard algebraic approach. This is achieved by attaching to each conformal vertex algebra a vector bundle with a flat connection on a formal disc, equipped with an intrinsic operation. The formal variable is restored when we choose a coordinate on the disc; the independence of the operation from the choice of coordinate follows from the fact that the group of changes of coordinates is an “internal symmetry” of the vertex algebra. Once we obtain a coordinate independent object, we can give a rigorous definition of the space of conformal blocks associated to a conformal vertex algebra and an algebraic curve X (see §5). From the physics point of view, conformal blocks give rise to “chiral correlation functions” on powers of X with singularities along the diagonals. As we vary the curve X and other data on X reflecting the internal symmetry of a vertex algebra (such as G–bundles), the corresponding spaces of coinvariants, which are the duals of the spaces of conformal blocks, combine into a sheaf on the relevant moduli space. This sheaf carries the structure of a (twisted) D–module, as explained in §6. One can gain new insights into the structure of moduli spaces from the study of these D–modules. For instance, one obtains a description of the formal deformation spaces of the complex structure or a G–bundle on a curve in terms of certain sheaves on the symmetric powers of the curve [BG, BD2, Gi]. Thus, vertex algebras appear as the local objects controlling deformations of curves with various extra structures. This raises the possibility that more exotic vertex algebras, such as W–algebras, also correspond to some still unknown moduli spaces. Finally, in §7 we discuss the relation between vertex algebras and the Beilinson–Drinfeld chiral algebras. We review briefly the description of chiral algebras as factorization alge- bras, i.e., sheaves on the Ran space of finite subsets of a curve, satisfying certain com- patibilities. Using this description, Beilinson and Drinfeld have introduced the concept of chiral homology, which can be thought of as a derived functor of the functor of coin- variants. The formalism of vertex and chiral algebras appears to be particularly suitable for the construction of the conjectural geometric Langlands correspondence between automorphic D–modules on the moduli space of G–bundles on a smooth projective curve X over C and flat LG–bundles on X, where G is a simple algebraic group and LG is the Langlands dual group (see [BD2]). The idea is to construct the automorphic D–modules corresponding to flat LG–bundles as the sheaves of coinvariants (or, more generally, chiral homology) of a 875-03 suitable vertex algebra. We mention below two such constructions (both due to Beilinson and Drinfeld): in one of them the relevant vertex algebra is associated to an affine Kac- Moody algebra of critical level (see §6.5), and in the other it is the chiral Hecke algebra (see §7). Another application of vertex algebras to algebraic geometry is the recent construction by Malikov, Schechtman and Vaintrob [MSV] of a sheaf of vertex superalgebras on an arbitrary smooth algebraic variety, called the chiral deRham complex, which is reviewed in §6.6. Vertex algebras form a vast and rapidly growing subject, and it is impossible to cover all major results (or even give a comprehensive bibliography) in one survey. For example, because of lack of space, I have not discussed such important topics as the theory of conformal algebras [K2, K3] and their chiral counterpart, Lie∗ algebras [BD2]; quantum deformations of vertex algebras [B3, EK, FR]; and the connection between vertex algebras and integrable systems. Most of the material presented below (note that §§4–6 contain previously unpublished results) is developed in the forthcoming book [FB]. I thank A. Beilinson for answering my questions about chiral algebras and D. Ben-Zvi for helpful comments on the draft of this paper. The support from the Packard Foundation and the NSF is gratefully acknowledged.

2. DEFINITION AND FIRST PROPERTIES OF VERTEX ALGEBRAS

2.1. Let R be a C–algebra. An R–valued formal power series (or formal distribution) in variables z1, z2, . . . , zn is an arbitrary infinite series

X X i1 in A(z1, . . . , zn) = ··· ai1,...,in z1 ··· zn , i1∈Z in∈Z ±1 ±1 where each ai1,...,in ∈ R. These series form a vector space denoted by R[[z1 , . . . , zn ]]. ±1 ±1 ±1 ±1 If P (z1, . . . , zn) ∈ R[[z1 , . . . , zn ]] and Q(w1, . . . , wm) ∈ R[[w1 , . . . , wm ]], then their ±1 ±1 ±1 ±1 product is a well-defined element of R[[z1 , . . . , zn , w1 , . . . , wm ]]. In general, a product of two formal power series in the same variables does not make sense, but the product of a formal power series by a polynomial (i.e., a series, such that ai1,...,in = 0 for all but finitely many indices) is always well-defined. ∞ Let V be a Z+–graded vector space V = ⊕n=0Vn with finite-dimensional homogeneous components. An endomorphism T of V is called homogeneous of degree n if T (Vm) ⊂

Vn+m. Denote by End V the vector space of linear endomorphisms of V , which are finite linear combinations of homogeneous endomorphisms. This is a Z–graded algebra. A field of conformal dimension ∆ ∈ Z+ is an End V –valued formal power series in z, X −j−∆ φ(z) = φjz j∈Z 875-04 where each φj is a homogeneous linear endomorphism of V of degree −j. Two fields φ(z) and ψ(z) are called mutually local if there exists N ∈ Z+, such that (1) (z − w)N [φ(z), ψ(w)] = 0 (as an element of End V [[z±1, w±1]]). Now we can formulate the axioms of vertex algebra.

Definition 2.1.— A vertex algebra is a collection of data: ∞ • space of states, a Z+–graded vector space V = ⊕n=0Vn, with dim(Vn) < ∞; • vacuum vector |0i ∈ V0; • shift operator T : V → V of degree one; −1 • vertex operation, a linear map Y (·, z): V → End V [[z, z ]] taking each A ∈ Vn to a field of conformal dimension n. These data are subject to the following axioms:

• (vacuum axiom) Y (|0i, z) = IdV . Furthermore, for any A ∈ V we have Y (A, z)|0i ∈ A + zV [[z]] (i.e., Y (A, z)|0i has a well–defined value at z = 0, which is equal to A).

• (translation axiom) For any A ∈ V , [T,Y (A, z)] = ∂zY (A, z) and T |0i = 0. • (locality axiom) All fields Y (A, z) are mutually local with each other.

Remark 2.2.— The locality axiom was first introduced in [DL], where it was shown that it can be used as a replacement for the original axioms of [B1, FLM] (see also [Li]). It is easy to adopt the above definition to the supercase (see, e.g., [K2]). Then V is a superspace, and the above structures and axioms should be modified appropriately; in particular, we need to replace the commutator by the supercommutator in the definition of locality. Then we obtain the definition of vertex superalgebra. We mostly consider below purely even vertex superalgebras, but general vertex superalgebras are very important; for instance, N = 2 superconformal vertex superalgebras appear in physical models relevant to mirror symmetry. The above conditions on V can be relaxed: it suffices to require that for any A ∈ V, v ∈

V , we have: An · v = 0 for n large enough. It is not necessary to require that dim Vn < ∞ and even that V is graded (in that case however one needs to be careful when dealing with dual spaces). We impose the above stronger conditions in order to simplify the exposition. It is straightforward to define homomorphisms between vertex algebras, vertex subal- gebras, ideals and quotients. If V1 and V2 are two vertex algebras, then V1 ⊗ V2 carries a natural vertex algebra structure.

2.2. Example: commutative vertex algebras.

Let V be a Z–graded commutative algebra (with finite-dimensional homogeneous com- ponents) with a unit and a derivation T of degree 1. Then V carries a canonical structure 875-05 of vertex algebra. Namely, we take the unit of V as the vacuum vector |0i, and define the operation Y by the formula X 1 Y (A, z) = m(T nA)zn, n! n≥0 where for B ∈ V , m(B) denotes the operator of multiplication by B on V . It is straight- forward to check that all axioms of vertex algebra are satisfied; in fact, the locality axiom is satisfied in a strong sense: for any A, B ∈ V , we have: [Y (A, z),Y (B, w)] = 0 (so we have N = 0 in formula (1)). Conversely, let V be a vertex algebra, in which locality holds in the strong sense (we call such a vertex algebra commutative). Then locality and vacuum axioms imply that Y (A, z) ∈ End V [[z]] for all A ∈ V (i.e., there are no terms with negative powers of z).

Denote by YA the endomorphism of V , which is the constant term of Y (A, z), and define a bilinear operation ◦ on V by setting A◦B = YA ·B. By construction, YAYB = YBYA for all A, B ∈ V . This implies commutativity and associativity of ◦ (see, e.g., [Li]). We obtain a commutative and associative product on V , which respects the Z–gradation. Furthermore, the vacuum vector |0i is a unit, and the operator T is a derivation with respect to this product. Thus, we see that the notion of commutative vertex algebra is equivalent to that of Z–graded commutative associative algebra with a unit and a derivation of degree 1.

Remark 2.3.— The operator T may be viewed as the generator of infinitesimal trans- lations on the formal additive group with coordinate z. Therefore a commutative vertex algebra is the same as a commutative algebra equipped with an action of the formal additive group. Thus one may think of general vertex algebras as meromorphic general- izations of commutative algebras with an action of the formal additive group. This point of view has been developed by Borcherds [B3], who showed that vertex algebras are “sin- gular commutative rings” in a certain category. He has also considered generalizations of vertex algebras, replacing the formal additive group by other (formal) groups or Hopf algebras.

2.3. Non-commutative example: Heisenberg vertex algebra.

Consider the space C((t)) of Laurent series in one variable as a commutative Lie algebra. We define the Heisenberg Lie algebra H as follows. As a vector space, it is the direct sum of the space of formal Laurent power series C((t)) and a one-dimensional space C1, with the commutation relations

(2) [f(t), g(t)] = − Res fdg1, [1, f(t)] = 0.

Here Res denotes the (−1)st Fourier coefficient of a Laurent series. Thus, H is a one- dimensional central extension of the commutative Lie algebra C((t)). Note that the relations (2) are independent of the choice of local coordinate t. Thus we may define a 875-06

Heisenberg Lie algebra canonically as a central extension of the space of functions on a punctured disc without a specific choice of formal coordinate t. n The Heisenberg Lie algebra H is topologically generated by the generators bn = t , n ∈ Z, and the central element 1, and the relations between them read

(3) [bn, bm] = nδn,−m1, [1, bn] = 0.

The subspace C[[t]] ⊕ 1 is a commutative Lie subalgebra of H. Let π be the H–module induced from the one-dimensional representation of C[[t]]⊕1, on which C[[t]] acts by 0 and 1 acts by 1. Equivalently, we may describe π as the polynomial algebra C[b−1, b−2,... ] ∂ with bn, n < 0, acting by multiplication, b0 acting by 0, and bn, n > 0, acting as n . ∂b−n The operators bn with n < 0 are known in this context as creation operators, since they “create the state bn from the vacuum 1”, while the operators bn with n ≥ 0 are the annihilation operators, repeated application of which will kill any vector in π. The module π is called the Fock representation of H. We wish to endow π with a structure of vertex algebra. This involves the following data (Definition 2.1): P • Z+ grading: deg bj1 . . . bjk = − i ji. • Vacuum vector: |0i = 1.

• The shift operator T defined by the rules: T · 1 = 0 and [T, bi] = −ibi−1. We now need to define the fields Y (A, z). To the vacuum vector |0i = 1, we are required to assign Y (|0i, z) = Id. The key definition is that of the field b(z) = Y (b−1, z), which generates π in an appropriate sense. Since deg(b−1) = 1, b(z) needs to have conformal dimension one. We set X −n−1 b(z) = bnz . n∈Z The Reconstruction Theorem stated below implies that once we have defined Y (b−1, z), there is (at most) a unique way to extend this definition to other vectors of π. This is not very surprising, since as a commutative algebra with derivation T , π is freely generated by b−1. Explicitly, the fields corresponding to other elements of π are constructed by the formula 1 n1−1 nk−1 (4) Y (b−n1 b−n2 . . . b−nk , z) = :∂z b(z) . . . ∂z b(z):. (n1 − 1)! ... (nk − 1)! The columns in the right hand side of the formula stand for the normally ordered product, which is defined as follows. First, let : bn1 . . . bnk : be the monomial obtained from bn1 . . . bnk by moving all bni with ni < 0 to the left of all bnj with nj ≥ 0 (in other words, moving all “creation operators” bn, n < 0, to the left and all “annihilation operators” bn, n ≤ 0, to the right). The important fact that makes this definition correct is that the operators bn with n < 0 (resp., n ≥ 0) commute with each other, hence it does not matter how we order the creation (resp., annihilation) operators among themselves (this 875-07 property does not hold for more general vertex algebras, and in those cases one needs to use a different definition of normally ordered product, which is given below). Now

m1 mk define :∂z b(z) . . . ∂z b(z): as the power series in z obtained from the ordinary product m1 mk ∂z b(z) . . . ∂z b(z) by replacing each term bn1 . . . bnk with :bn1 . . . bnk :.

Proposition 2.4.— The Fock representation π with the structure given above satisfies the axioms of vertex algebra.

The most difficult axiom to check is locality. The Reconstruction Theorem stated below allows us to reduce it to verifying locality property for the “generating” field b(z). This is straightforward from the commutation relations (3):

X −n−1 −m−1 X −n−1 n−1 [b(z), b(w)] = [bn, bm]z w = [bn, b−n]z w n,m∈Z n∈Z X −n−1 n−1 = nz w = ∂wδ(z − w), n∈Z X where we use the notation δ(z − w) = wmz−m−1. m∈Z Now δ(z − w) has the property that (z − w)δ(z − w) = 0. In fact, the annihilator of the operator of multiplication by (z − w) in R[[z±1, w±1]] equals R[[w±1]] · δ(z − w) (note that the product of δ(z − w) with any formal power series in z or in w is well- defined). More generally, the annihilator of the operator of multiplication by (z − w)N in ±1 ±1 N−1 ±1 n R[[z , w ]] equals ⊕n=0 R[[w ]] · ∂wδ(z − w) (see [K1]). This implies in particular that (z − w)2[b(z), b(w)] = 0, and hence the field b(z) is local with itself. Proposition 2.4 now follows from the Reconstruction Theorem below.

2.4. Reconstruction Theorem.

We state a general result, which provides a “generators–and–relations” approach to the construction of vertex algebras. Let V be a Z+–graded vector space, |0i ∈ V0 a non-zero vector, and T a degree 1 endomorphism of V . Let S be a countable ordered set and α α {a }α∈S be a collection of homogeneous vectors in V , with a of degree ∆α. Suppose we are also given fields

α X α −n−∆α a (z) = anz n∈Z on V , such that the following hold: • For all α, aα(z)|0i ∈ aα + zV [[z]]; α α • T |0i = 0 and [T, a (z)] = ∂za (z) for all α; • All fields aα(z) are mutually local; • V is spanned by lexicographically ordered monomials

aα1 . . . aαm |0i −∆α1 −j1 −∆αm −jm 875-08

(including |0i), where j1 ≥ j2 ≥ ... ≥ jm ≥ 0, and if ji = ji+1, then αi ≥ αi+1 with respect to the order on the set S.

Theorem 2.5 ([FKRW]).— Under the above assumptions, the above data together with the assignment 1 (5) Y (aα1 . . . aαm |0i, z) = :∂j1 aα1 (z) . . . ∂jm aαm (z):, −∆α −j1 −∆αm −jm z z 1 j1! . . . jm! define a vertex algebra structure on V .

Here we use the following general definition of the normally ordered product of fields.

Let φ(z), ψ(w) be two fields of respective conformal dimensions ∆φ, ∆ψ and Fourier coefficients φn, ψn. The normally ordered product of φ(z) and ψ(z) is by definition the formal power series  

X X X −n−∆φ−∆ψ :φ(z)ψ(z): =  φmψn−m + ψn−mφm z . n∈Z m≤−∆φ m>−∆φ

This is a field of conformal dimension ∆φ + ∆ψ. The normal ordering of more than two fields is defined recursively from right to left, so that by definition :A(z)B(z)C(z): = :A(z)(:B(z)C(z):):. It is easy to see that in the case of the Heisenberg vertex algebra π this definition of normal ordering coincides with the one given in §2.3.

2.5. The meaning of locality. The product φ(z)ψ(w) of two fields is a well–defined End V –valued formal power series in z±1 and w±1. Given v ∈ V and ϕ ∈ V ∗, consider the matrix coefficient

±1 ±1 hϕ, φ(z)ψ(w)vi ∈ C[[z , w ]].

Since Vn = 0 for n < 0, we find by degree considerations that it belongs to C((z))((w)), the space of formal Laurent series in w, whose coefficients are formal Laurent series in z. Likewise, hϕ, ψ(w)φ(z)vi belongs to C((w))((z)). As we have seen in §2.2, the condition that the fields φ(z) and ψ(w) literally commute is too strong, and it essentially keeps us in the realm of commutative algebra. However, there is a natural way to relax this condition, which leads to the more general notion of locality. Let C((z, w)) be the field of fractions of the ring C[[z, w]]; its elements may be viewed as meromorphic functions in two formal variables. We have natural embeddings

(6) C((z))((w)) ←− C((z, w)) −→ C((w))((z)). which are simply the inclusions of C((z, w)) into its completions in two different topologies, corresponding to the z and w axes. For example, the images of 1/(z − w) ∈ C((z, w)) in C((z))((w)) and in C((w))((z)) are equal to X 1 X δ(z − w) = wnz−n−1, −δ(z − w) = − wnz−n−1, − + z n≥0 n<0 875-09 respectively. Now we can relax the condition of commutativity of two fields by requiring that for any v ∈ V and ϕ ∈ V ∗, the matrix elements hϕ, φ(z)ψ(w)vi and hϕ, ψ(w)φ(z)vi are the images of the same element fv,ϕ of C((z, w)) in C((z))((w)) and C((w))((z)), respectively. In the case of vertex algebras, we additionally require that

−1 −1 −1 fv,ϕ ∈ C[[z, w]][z , w , (z − w) ] ∗ for all v ∈ V, ϕ ∈ V , and that the order of pole of fv,ϕ is universally bounded, i.e., there N −1 −1 exists N ∈ Z+, such that (z −w) fv,ϕ ∈ C[[z, w]][z , w ] for all v, ϕ. The last condition is equivalent to the condition of locality given by formula (1). From the analytic point of view, locality of the fields φ(z) and φ(w) means the following. ∗ Suppose that ϕ is an element of the restricted dual space ⊕n≥0Vn . Then hϕ, φ(z)ψ(w)vi converges in the domain |z| > |w| whereas hϕ, ψ(w)φ(z)vi converges in the domain |w| >

|z|, and both can be analytically continued to the same meromorphic function fv,ϕ(z, w) ∈ C[z, w][z−1, w−1, (z − w)−1]. Then their commutator (considered as a distribution) is the difference between boundary values of meromorphic functions, and hence is a delta-like distribution supported on the diagonal. In the simplest case, this amounts to the formula

δ(z − w) = δ(z − w)− + δ(z − w)+, which is a version of the Sokhotsky–Plemelj formula well-known in complex analysis.

2.6. Associativity. Now we state the “associativity” property of vertex algebras. Consider the V –valued formal power series

X −n−∆A Y (Y (A, z − w)B, w)C = Y (An · B, w)C(z − w) n∈Z in w and z − w. By degree reasons, this is an element of V ((w))((z − w)). By locality, Y (A, z)Y (B, w)C is the expansion in V ((z))((w)) of an element of V [[z, w]][z−1, w−1, (z − w)−1], which we can map to V ((w))((z − w)) sending z−1 to (w + (z − w))−1 considered as a power series in positive powers of (z − w)/w.

Proposition 2.6 ([FHL, K2]).— Any vertex algebra V satisfies the following associa- tivity property: for any A, B, C ∈ V we have the equality in V ((w))((z − w)) (7) Y (A, z)Y (B, w)C = Y (Y (A, z − w)B, w)C.

N One can also show that for large enough N ∈ Z+, z Y (Y (A, z − w)B, w)C belongs to V [[z, w]][w−1, (z − w)−1], and its image in V [[z±1, w±1]], obtained by expanding (z − w)−1 in positive powers of w/z, equals zN Y (A, z)Y (B, w)C. Formula (7) is called the operator product expansion (OPE). From the physics point of view, it manifests the important property in quantum field theory that the product of two fields at nearby points can be expanded in terms of other fields and the small parameter 875-10 z − w. From the analytic point of view, this formula expresses the fact that the matrix elements of the left and right hand sides of the formula, well-defined in the appropriate domains, can be analytically continued to the same rational functions in z, w, with poles only at z = 0, w = 0, and z = w. To simplify formulas we will often drop the vector C in formula (7). For example, in the case of Heisenberg algebra, we obtain: 1 X 1 1 (8) b(z)b(w) = + : ∂nb(w)b(w):(z − w)n = + : b(z)b(w): . (z − w)2 n! w (z − w)2 n≥0 Using the Cauchy formula, one can easily extract the commutation relations between the Fourier coefficients of the fields Y (A, z) and Y (B, w) from the singular (at z = w) terms of their OPE. The result is ! X m + ∆A − 1 (9) [Am,Bk] = (An · B)m+k. n + ∆A − 1 n>−∆A

2.7. Correlation functions.

Formula (7) and the locality property have the following “multi–point” generalization. Let V ∗ be the space of all linear functionals on V .

∗ Proposition 2.7 ([FHL]).— Let A1,...,An ∈ V . For any v ∈ V, ϕ ∈ V , and any permutation σ on n elements, the formal power series in z1, . . . , zn,
(10) ϕ Y (Aσ(1), zσ(1)) ...Y (Aσ(n), zσ(n))|0i

σ is the expansion in C((zσ(1))) ... ((zσ(n))) of an element fA1,...,An (z1, . . . , zn) of −1 C[[z1, . . . , zn]][(zi − zj) ]i6=j, which satisfies the following properties: it does not depend on σ (so we suppress it in the notation);

fA ,...,A (z1, . . . , zn) = f (zj, z1,..., zi,..., zj, . . . , zn) 1 n Y (Ai,zi−zj )Aj ,A1,...,Aci,...,Acj ,...,An b b for all i 6= j; and ∂zi fA1,...,An (z1, . . . , zn) = fA1,...,T Ai,...,An (z1,...,An).

Thus, to each ϕ ∈ V ∗ we can attach a collection of matrix elements (10), the “n–point −1 functions” on Spec C[[z1, . . . , zn]][(zi − zj) ]i6=j. They satisfy a symmetry condition, a “bootstrap” condition (which describes the behavior of the n–point function near the diagonals in terms of (n − 1)–point functions) and a “horizontality” condition. We obtain ∗ a linear map from V to the vector space Fn of all collections {fA1,...,Am (z1, . . . , zm),Ai ∈ n V }m=1 satisfying the above conditions. This map is actually an isomorphism for each ∗ n n ≥ 1. The inverse map Fn → V takes {fA1,...,Am (z1, . . . , zm),Ai ∈ V }m=1 ∈ Fn to the functional ϕ on V , defined by the formula ϕ(A) = fA(0). Thus, we obtain a “functional realization” of V ∗. In the case when V is generated by fields such that the singular terms in their OPEs are linear combinations of the same fields and their derivatives, we 875-11 can simplify this functional realization by considering only the n–point functions of the generating fields. For example, in the case of the Heisenberg vertex algebra we consider for each ϕ ∈ π∗ the n–point functions

(11) ωn(z1, . . . , zn) = ϕ(b(z1) . . . b(zn)|0i). By Proposition 2.7 and the OPE (8), these functions are symmetric and satisfy the boot- strap condition   ωr−2(z1,..., zbi,..., zbj, . . . , zn) (12) ωn(z1, . . . , zn) = 2 + regular . (zi − zj)

Let Ω∞ be the vector space of infinite collections (ωn)n≥0, where

−1 ωn ∈ C[[z1, . . . , zn]][(zi − zj) ]i6=j ∗ satisfy the above conditions. Using the functions (11), we obtain a map π → Ω∞. One can show that this map is an isomorphism.

3. MORE EXAMPLES

3.1. Affine Kac-Moody algebras.

Let g be a simple Lie algebra over C. Consider the formal loop algebra Lg = g((t)) with the obvious commutator. The affine algebra bg is defined as a central extension of Lg. As a vector space, bg = Lg ⊕ CK, and the commutation relations read: [K, ·] = 0 and

(13) [A ⊗ f(t),B ⊗ g(t)] = [A, B] ⊗ f(t)g(t) + (Rest=0 fdg(A, B))K Here (·, ·) is an invariant bilinear form on g (such a form is unique up to a scalar multiple; we normalize it as in [K1] by the requirement that (αmax, αmax) = 2).

Consider the Lie subalgebra g[[t]] of Lg. If f, g ∈ C[[t]], then Rest=0 fdg = 0. Hence g[[t]] is a Lie subalgebra of bg. Let Ck be the one–dimensional representation of g[[t]]⊕CK, on which g[[t]] acts by 0 and K acts by the scalar k ∈ C. We define the vacuum repre- bg sentation of g of level k as the representation induced from k: Vk(g) = Ind k. b C g[[t]]⊕CK C a a a n a Let {J }a=1,... dim g be a basis of g. Denote Jn = J ⊗ t ∈ Lg. Then Jn, n ∈ Z, and K form a (topological) basis for bg. By the Poincar´e–Birkhoff–Witt theorem, Vk(g) has a basis a1 am of lexicographically ordered monomials of the form Jn1 ...Jnm vk, where vk is the image of 1 ∈ Ck in Vk(g), and all ni < 0. We are now in the situation of the Reconstruction Theorem, and hence we obtain a vertex algebra structure on Vk(g), such that

a a X a −n−1 Y (J−1vk, z) = J (z) := Jnz . n∈Z The fields corresponding to other monomials are obtained by formula (5). Explicit com- putation shows that the fields J a(z) (and hence all other fields) are mutually local. 875-12

3.2. Virasoro algebra.

∂ The Virasoro algebra V ir is a central extension of the Lie algebra Der C((t)) = C((t)) ∂t . n+1 It has (topological) basis elements Ln = −t , n ∈ Z, and C satisfying the relations that C is central and n3 − n [L ,L ] = (n − m)L + δ C n m n+m 12 n,−m (a coordinate independent version is given in §5.2). These relations imply that the Lie ∂ algebra Der C[[t]] = C[[t]] ∂t generated by Ln, n ≥ 0, is a Lie subalgebra of V ir. Consider the induced representation Vir = IndV ir , where is a one–dimensional repre- c Der C[[t]]⊕CC Cc Cc sentation, on which C acts as c and Der C[[t]] acts by zero. By the Poincar´e–Birkhoff–Witt theorem, Virc has a basis consisting of monomials of the form Lj1 ...Ljm vc, j1 ≤ j2 ≤ jm ≤ −2, where vc is the image of 1 ∈ Cc in the induced representation. By Reconstruction Theorem, we obtain a vertex algebra structure on Virc, such that

X −n−2 Y (L−2vc, z) = T (z) := Lnz . n∈Z

Note that the translation operator is equal to L−1. The Lie algebra Der O generates infinitesimal changes of coordinates. As we will see in §4, it is important to have this Lie algebra (and even better, the whole Virasoro algebra) act on a given vertex algebra by “internal symmetries”. This property is formalized by the following definition.

Definition 3.1.— A vertex algebra V is called conformal, of central charge c ∈ C, if V contains a non-zero conformal vector ω ∈ V2, such that the corresponding vertex operator P −n−2 1 Y (ω, z) = Lnz satisfies: L−1 = T , L0|V = n Id, and L2ω = c|0i. n∈Z n 2

These conditions imply that the operators Ln, n ∈ Z, satisfy the commutation relations of the Virasoro algebra with central charge c. We also obtain a non-trivial homomorphism

Virc → V , sending L−2vc to ω. 1 2 For example, the vector 2 b−1 + λb−2 is a conformal vector in π for any λ ∈ C. The 2 corresponding central charge equals 1 − 12λ . The Kac-Moody vertex algebra Vk(g) is conformal if k 6= −h∨, where h∨ is the dual Coxeter number of g. The conformal vector is 1 P a 2 a given by the Sugawara formula 2(k+h∨) a(J−1) vk, where {J } is an orthonormal basis of ∨ g. Thus, each bg–module from the category O of level k 6= −h is automatically a module over the Virasoro algebra.

3.3. Boson–fermion correspondence.

Let C be the Clifford algebra associated to the vector space C((t)) ⊕ C((t))dt equipped with the inner product induced by the residue pairing. It has topological generators n ∗ n−1 ψn = t , ψn = t dt, n ∈ Z, satisfying the anti-commutation relations ∗ ∗ ∗ (14) [ψn, ψm]+ = [ψn, ψm]+ = 0, [ψn, ψm]+ = δn,−m. 875-13

Denote by V the fermionic Fock representation of C, generated by vector |0i, such that ∗ ψn|0i = 0, n ≥ 0, ψn|0i = 0, n > 0. This is a vertex superalgebra with

X −n−1 ∗ ∗ X ∗ −n Y (ψ−1|0i, z) = ψ(z) := ψnz ,Y (ψ0|0i, z) = ψ (z) := ψnz . n∈Z n∈Z Boson–fermion correspondence establishes an isomorphism between V and a vertex superalgebra built out of the Heisenberg algebra H from §2.3. For λ ∈ C, let πλ be the H–module generated by a vector 1λ, such that bn · 1λ = λδn,01λ, n ≥ 0. For N ∈ Z>0, √ √ set V = ⊕m∈ π . This is a vertex algebra (resp., superalgebra) for any even N NZ Z m N (resp., odd N), which contains π0 = π as a vertex subalgebra. The gradation is given 2 by the formulas deg bn = −n, deg 1λ = λ /2 (note that it can be half-integral). The 1 P translation operator is T = bnb−1−n, so T · 1λ = λb−11λ. In order to define the 2 n∈Z vertex operation on V√ , it suffices to define the fields Y (1 √ , z). They are determined NZ m N by the identity

∂zY (1λ, z) = Y (T · 1λ, z) = λ : b(z)Y (1λ, z): . Explicitly, ! ! X bn X bn Y (1 , z) = S zλb0 exp −λ z−n exp −λ z−n , λ λ n n n<0 n>0 where Sλ : πµ → πµ+λ is the shift operator, Sλ · 1µ = 1µ+λ, [Sλ, bn] = 0, n 6= 0. V The boson–fermion correspondence is an isomorphism of vertex superalgebras ' VZ, ∗ which maps ψ(z) to Y (1−1, z) and ψ (z) to Y (11, z). For more details, see, e.g., [K2].

3.4. Rational vertex algebras.

Rational vertex algebras constitute an important class of vertex algebras, which are particularly relevant to conformal field theory [dFMS]. In order to define them, we first need to give the definition of a module over a vertex algebra. Let V be a vertex algebra. A vector space M is called a V –module if it is equipped with the following data:

• gradation: M = ⊕n∈Z+ Mn; −1 • operation YM : V → End M[[z, z ]], which assigns to each A ∈ Vn a field YM (A, z) of conformal dimension n on M; subject to the following conditions:

• YM (|0i, z) = IdM ;

• For all A, B ∈ V and C ∈ M, the series YM (A, z)YM (B, w)C is the expansion of an −1 −1 −1 element of M[[z, w]][z , w , (z − w) ] in M((z))((w)), and YM (A, z)YM (B, w)C

= YM (Y (A, z − w)B, w)C, in the sense of Proposition 2.6. 875-14

A conformal vertex algebra V (see Definition 3.1) is called rational if every V –module is completely reducible. It is shown in [DLM] that this condition implies that V has finitely many inequivalent simple modules, and the graded components of each simple V –module M are finite-dimensional. Furthermore, the gradation operator on M coincides with the M Virasoro operator L0 up to a shift. Hence we can attach to a simple V –module M its LM −c/24 2πiτ character ch M = TrM q 0 , where c is the central charge of V . Set q = e . Zhu [Z1] has shown that if V satisfies a certain finiteness condition, then the characters give rise to holomorphic functions in τ on the upper-half plane. Moreover, he proved the following remarkable Theorem 3.2.— Let V be a rational vertex algebra satisfying Zhu’s finiteness condition, and {M1,...,Mn} be the set of all inequivalent simple V –modules. Then the vector space spanned by ch Mi, i = 1, . . . , n, is invariant under the action of SL2(Z). This result has the following heuristic explanation. To each vertex algebra we can attach the sheaves of coinvariants on the moduli spaces of curves (see §6). It is expected that the sheaves associated to a rational vertex algebra satisfying Zhu’s condition are vector bundles with projectively flat connection. The characters of simple V –modules should form the basis of the space of horizontal sections of the corresponding bundle on the moduli of elliptic curves near τ = ∞. The SL2(Z) action is just the monodromy action on these sections. Here are some examples of rational vertex algebras. (1) Let L be an even positive definite lattice in a real vector space W . One can attach √ to it a vertex algebra VL in the same way as in §3.3 for L = NZ. Namely, we define the Heisenberg Lie algebra associated to WC as the Kac-Moody type central extension of the commutative Lie algebra WC((t)). Its Fock representation πW is a vertex algebra. The vertex structure on πW can be extended to VL = πW ⊗ C[L], where C[L] is the group algebra of L [B1, FLM]. Then VL is rational, and its inequivalent simple modules are parameterized by L0/L where L0 is the dual lattice [D1]. The corresponding characters are theta-functions. The vertex algebra VL is the chiral symmetry algebra of the free bosonic conformal field theory compactified on the torus W/L.

Note that if we take as L an arbitrary integral lattice, then VL is a vertex superalgebra. (2) Let g be a simple Lie algebra and k ∈ Z+. Let Lk(g) be the irreducible quotient of the bg–module Vk(g). This is a rational vertex algebra, whose modules are integrable representations of bg of level k [FZ]. The corresponding conformal field theory is the Wess-Zumino-Witten model (see [dFMS]).

(3) Let Lc(p,q) be the irreducible quotient of Virc(p,q) (as a module over the Virasoro algebra), where c(p, q) = 1 − 6(p − q)2/pq, p, q > 1, (p, q) = 1. This is a rational vertex algebra [Wa], whose simple modules form the “minimal model” of conformal field theory defined by Belavin, Polyakov, and Zamolodchikov [BPZ] (see also [dFMS]). (4) The Moonshine Module vertex algebra V \ (see below). 875-15

Zhu [Z1] has attached to each vertex algebra V an associative algebra A(V ), such that simple V –modules are in one-to-one correspondence with simple A(V )–modules.

3.5. Orbifolds and the Monster.

Let G be a group of automorphisms of a vertex algebra V . Then the space V G of G– invariants in V is a vertex subalgebra of V . When G is a finite group, orbifold theory allows one to construct V G–modules from twisted V –modules. For a finite order automorphism g of V , one defines a g–twisted V –module by modifying the above axioms of V –module −1 in such a way that YM (A, z) has monodromy λ around the origin, if g · A = λv (see [DLM]). Then conjecturally all V G–modules can be obtained as V G–submodules of g– twisted V –modules for g ∈ G. Now suppose that V is a holomorphic vertex algebra, i.e., rational with a unique simple module, namely itself. Let G be a cyclic group of automorphisms of V of order n generated by g. Then the following pattern is expected to hold (see [D2]): (1) for each h ∈ G there is a unique simple h–twisted V –module Vh; (2) Vh has a natural G–action, so we can n−1 2πik/n write Vh = ⊕i=0 Vh(i), where Vh(i) = {v ∈ Vh|g · v = e v}; then each Vh(i) is a simple G G V –module, and these are all simple V –modules; (3) The vector space Ve = ⊕h∈GVh(0) carries a canonical vertex algebra structure, which is holomorphic. A spectacular example is the construction of the Moonshine Module vertex algebra V \ by I. Frenkel, Lepowsky and Meurman [FLM]. In this case V is the vertex algebra

VΛ associated to the Leech lattice Λ (it is self-dual, hence VΛ is holomorphic), and g \ is constructed from the involution −1 on Λ. Then V = VΛ(0) ⊕ VΛ,g(0) is a vertex algebra, whose group of automorphisms is the Monster group [FLM]. Moreover, V \ is holomorphic [D2]. Conjecturally, V \ is the unique holomorphic vertex algebra V with central charge 24, such that V1 = 0. Its character is the modular function j(τ) − 744. More generally, for each element x of the Monster group, consider the Thompson series

L0−1 TrV \ xq . The Conway–Norton conjecture, in the formulation of [FLM], states that these are Hauptmoduls for genus zero subgroups of SL2(R). This was first proved in [FLM] in the case when x commutes with a certain involution, and later in the general case (by a different method) by R. Borcherds [B2].

3.6. Coset construction.

Let V be a vertex algebra, and W a vector subspace of V . Denote by C(V,W ) the vector subspace of V spanned by vectors v such that Y (A, z) · v ∈ V [[z]] for all A ∈ W . Then C(V,W ) is a vertex subalgebra of V , which is called the coset vertex algebra of the pair (V,W ). In the case W = V , the vertex algebra C(V,V ) is commutative, and is called the center of V . An example of a coset vertex algebra is provided by the Goddard–Kent–

Olive construction [GKO], which identifies C(L1(sl2) ⊗ Lk(sl2),Lk+1(sl2)) with Lc(k+2,k+3) (see §3.4). For other examples, see [dFMS]. 875-16

3.7. BRST construction and W–algebras.

For A ∈ V , let y(A) = Res Y (A, z). Then associativity implies: [y(A),Y (B, z)] = Y (y(A) · B, z), and so y(A) is an infinitesimal automorphism of V . Now suppose that V • is a vertex (super)algebra with an additional Z–gradation and A ∈ V 1 is such that y(A)2 = 0. Then (V •, y(A)) is a complex, and its cohomology is a graded vertex (su- per)algebra. Important examples of such complexes are provided by topological vertex superalgebras introduced by Lian and Zuckerman [LZ]. In this case the cohomology is a graded commutative vertex algebra, but it has an additional structure of Batalin- Vilkovisky algebra. Another example is the BRST complex of quantum hamiltonian reduction. We il- lustrate it in the case of quantum Drinfeld-Sokolov reduction [FF3], which leads to the definition of W–algebras. Let g be a simple Lie algebra of rank `, and n its upper nilpotent subalgebra. Let C be the Clifford algebra associated to the vector space n((t)) ⊕ n∗((t))dt equipped with the inner product induced by the residue pairing. It has topological generators n ∗ ∗ n−1 ψα,n = eα ⊗ t , ψα,n = eα ⊗ t dt, α ∈ ∆+, n ∈ Z, satisfying the anti-commutation V relations (14). Let n be its Fock representation, generated by vector |0i, such that ∗ ψα,n|0i = 0, n ≥ 0, ψα,n|0i = 0, n > 0. This is a vertex superalgebra which is the tensor product of several copies of the vertex superalgebra V from §3.3. Introduce an additional V ∗ Z–gradation on C and n by setting deg ψα,n = − deg ψα,n = 1, deg |0i = 0. • V• Now consider the vertex superalgebra Ck (g) = Vk(g) ⊗ n. It carries a canonical differential dst of degree 1, the standard differential of semi-infinite cohomology of n((t)) • with coefficients in Vk [Fe1, FGZ]; it is equal to the residue of a field from Ck (g) (see [FF3]). Let χ = P` Res ψ∗ (z) be the Drinfeld-Sokolov character [DS] of n((t)). Then i=1 αi • • d = dst + χ is a differential on Ck (g), and the cohomology Hk (g) of this differential is a vertex superalgebra.

0 Theorem 3.3.— Hk (g) is a vertex algebra generated by elements Wi of degrees di +1, i = 1, . . . , `, where di is the ith exponent of g (in the sense of the Reconstruction Theorem), i and Hk(g) = 0, i 6= 0.

This theorem is proved in [FF3, FF4] for generic k and in [dBT] for all k. The vertex 0 algebra Hk (g) is called the W–algebra associated to g and denoted by Wk(g). We have: 2 Wk(sl2) = Virc(k), where c(k) = 1 − 6(k + 1) /(k + 2). For g = sl3, the W–algebra was

first constructed by Zamolodchikov, and for g = slN by Fateev and Lukyanov [FL]. ∨ Since Vk(g) is conformal for k 6= −h (see §3.2), Wk(g) is also conformal, with W1 playing the role of conformal vector. On the other hand, W−h∨ (g) is a commutative vertex algebra, which is isomorphic to the center of V−h∨ (g) [FF3] (see §6.5). The simple ∨ quotient of Wk(g) for k = −h + p/q, where p, q are relatively prime integers greater than or equal to h∨, is believed to be a rational vertex algebra. Moreover, if g is simply-laced 875-17 and q = p + 1, this vertex algebra is conjecturally isomorphic to the coset algebra of

(L1(g) ⊗ Lp−2(g),Lp−1(g)), see [FKW]. V• For any Vk(g)–module M, the cohomology of the complex (M ⊗ n, d) is a Wk(g)– module. This defines a functor, which was studied in [FKW]. L L The W–algebras exhibit a remarkable duality: Wk(g) ' Wk0 ( g), where g is the Langlands dual Lie algebra to g and (k + h∨)r∨ = (k0 + Lh∨)−1 (here r∨ denotes the maximal number of edges connecting two vertices of the Dynkin diagram of g), see [FF3]. In the limit k → −h∨ it becomes the isomorphism of Theorem 6.3.

4. COORDINATE INDEPENDENT DESCRIPTION OF VERTEX ALGEBRA STRUCTURE

Up to this point, we have discussed vertex algebras in the language of formal power series. The vertex operation is a map Y : V → End V [[z, z−1]], or, equivalently, an ∗ element of the completed tensor product V ((z))⊗b End V . Thus, we can view Y as an End V –valued section of a vector bundle over the punctured disc D× = Spec C((z)) with fiber V ∗. The question is whether one can define this bundle in such a way that this section is canonical, i.e., independent of the choice of z. In order to do that, we need a precise description of the transformation properties of Y . Once we understand what type of geometric object Y is, we will be able to give a global geometric meaning to vertex operators on arbitrary curves.

4.1. The group Aut O.

Denote by O the complete topological C–algebra C[[z]], and let Aut O be the group of continuous automorphisms of O. Such an automorphism is determined by its action on the generator z ∈ C[[z]]. Thus as a set this group consists of elements ρ(z) ∈ C[[z]] of the form 2 × ρ1z + ρ2z + ..., with ρ1 ∈ C , endowed with the composition law (ρ ∗ µ)(z) = µ(ρ(z)). It is easy to see that Aut O is a proalgebraic group, lim Aut [[z]]/(zn). It is the semi-direct ←− C product GmnAut+ O, where Aut+ O is the subgroup that consists of the transformations of 2 the form z → z+a2z +··· , and Gm is the group of rescalings z → az, a 6= 0. Furthermore, Aut+ O is a prounipotent group. The Lie algebra of Aut O is Der0 O = zC[[z]]∂z, which is 2 a semi–direct product of C∂z and Der+ O = z C[[z]]∂z. Any representation of Der0 O, on which z∂z is diagonalizable with integral eigenvalues and Der+ O acts locally nilpotently, can be exponentiated to a representation of Aut O. Given a vertex algebra and a field Y (A, z), it makes sense to consider a new field Y (A, ρ(z)). We now seek an action of Aut O on V , A 7→ R(ρ) · A, such that Y (A, ρ(z)) is related to Y (R(ρ)A, z) in some reasonable way. In the theory of vertex algebras, actions of Aut O usually arise from the action of the Virasoro algebra. Namely, let V be a conformal vertex algebra (see Definition 3.1). Then the Fourier coefficients Ln of Y (ω, z) satisfy 875-18 the commutation relations of the Virasoro algebra with central charge c. The operators n+1 Ln, n ≥ 0, then define an action of the Lie algebra Der+ O (Ln corresponds to −t ∂t). It follows from the axioms of vertex algebra that this action can be exponentiated to an action of Aut O. For f(z) ∈ Aut O, denote by R(f): V → V the corresponding operator. X n+1 X Given a vector field v = vnz ∂z, we assign to it the operator v = − vnLn. n≥−1 n≥−1 From formula (9) we obtain: X 1 (15) [v,Y (A, w)] = − (∂m+1v(w))Y (L A, w). (m + 1)! w m m≥−1 By “exponentiating” this formula, we obtain the following identity, due to Y.-Z. Huang [Hu]:

−1 −1 (16) Y (A, t) = R(ρ)Y (R(ρt) A, ρ(t))R(ρ) , for any ρ ∈ Aut C[[z]] (here ρt(z) = ρ(t + z) − ρ(t), considered as formal power series in z with coefficients in C[[t]]).

4.2. Vertex algebra bundle.

Now we can give a coordinate-free description of the operation Y . Let X be a smooth complex curve. Given a point x ∈ X, denote by Ox the completion of the local ring of x, by mx its maximal ideal, and by Kx the field of fractions of Ox. A formal coordinate tx at x is by definition a topological generator of mx. Consider the set of pairs (x, tx), where x ∈ X and tx is a formal coordinate at x. This is the set of points of a scheme Xb of infinite type, which is a principal Aut O–bundle over X. Its fiber at x ∈ X is the Aut O–torsor Lx of all formal coordinates at x. Given a finite-dimensional Aut O–module V , let V = Xb × V be the vector bundle associated to V and Xb. The fiber of V at x ∈ X Aut O is the Lx–twist of V , Vx = Lx × V . Aut O i A conformal vertex algebra is an Aut O–module, which has a filtration V≤i := ⊕k=0Vk by finite-dimensional Aut O–submodules. We obtain the directed system (V≤i) of the corresponding vector bundles of finite rank on X and embeddings V≤i ,→ V≤j, i ≤ j. We will denote this system simply by V, thinking of it as an inductive limit of bundles of ∗ ∗ finite rank. Likewise, by V we will understand the inverse system of bundles (V≤i) and ∗ i ∗ surjections (V≤j)  (V ) , thinking of it as a projective limit of bundles of finite rank. The Aut O–bundle Xb on X above carries an action of the Lie algebra Der O, which is compatible with the Aut O–action and simply transitive, i.e., Der O⊗O ' Θ . Since b Xb Xb V is a Der O–module, by the general construction of §6.1 we obtain a flat connection

∇ : V → V ⊗ Ω on V. Locally, ∇ can be written as d + L−1 ⊗ dz, where d the deRham differential. The connection ∇ on V gives us a connection ∇∗ on the dual bundle V∗.

To be precise, V is not an inductive limit of flat bundles, since ∇(V≤i) ⊂ Vi+1 ⊗ Ω, and 875-19 likewise, V∗ is not a projective limit of flat bundles (it is a D–module on X, which is not quasicoherent as an O–module). ∗ × Now let us restrict V to the punctured disc Dx = Spec Kx around x ∈ X. We want to ∗ × define an End Vx–valued meromorphic section Yx of V on Dx . Pick a formal coordinate z at x. With this choice, identify Vx with V and trivialize V|Dx . We define our section 0 ∗ Yx in this trivialization through its matrix coefficients: to each triple c ∈ V , c ∈ V and a section F : Dx → V we need to assign a meromorphic function on the disc, which is C–linear in c and c0 and O–linear in F . It suffices to assign such a function to triples c, c0,F , where F is the constant section equal to A ∈ V with respect to our trivialization. We set this function to be equal to hc0|Y (A, z)|ci.

Theorem 4.1.— The section Yx defined this way is canonical, i.e., independent of the ∗ choice of formal coordinate z at x. Moreover, Yx is horizontal: ∇ Yx = 0.

Thus, we see that a conformal vertex algebra gives rise to a vector bundle on any smooth curve and a canonical horizontal section of the restriction of this bundle to the neighborhood of each point with values in the endomorphisms of the corresponding twist of V . The fact that this section is canonical is equivalent to formula (16). Flatness follows from the formula ∂zY (B, z) = Y (L−1B, z), which holds in any vertex algebra.

Remark 4.2.— Let Vn be the n–fold external tensor power of V. Consider the n–point functions (10). According to Proposition 2.7, these are elements of C[[z1, . . . , zn]][(zi − −1 n n zj) ]i6=j. Trivializing the restriction of V to Dx = Spec C[[z1, . . . , zn]] using the coor- ∗ n ∗  n dinate z, we obtain a Vx–valued section of (V ) |Dx with poles on the diagonals. Then this section is canonical, i.e., independent of the choice of z, and horizontal.

4.3. Examples. If W is an Aut O–stable subspace of V , it gives rise to a subbundle W of V. The ∗ universal section Y then gives rise to an End V –valued section of W | × . This way one x x Dx obtains many familiar geometric objects.

Suppose A ∈ V∆ satisfies Ln · A = 0,L0A = ∆A. Then subspace CA ⊂ V is a one- dimensional Aut O submodule of V . The line bundle associated to this module is nothing −∆ but the line bundle Ω of ∆–differentials on X. Thus we obtain a line subbundle LA −1 ∆ of V. Our section Yx therefore gives us an End Vx–valued section of LA = Ω . In other ∆ words, the End Vx–valued ∆–differential Y (A, z)(dz) does not depend on the choice of formal coordinate z at x. Next, consider the Heisenberg vertex algebra V = π, with conformal structure given by 1 2 the vector 2 b−1 + b−2. For each smooth curve X, π gives rise to a vector bundle that we denote by Π. The first piece π0 of our filtration on π = C[bn]n<0 is the one-dimensional subspace spanned by the vacuum vector 1. This is a trivial representation of the group

Aut O. Hence it gives rise to a trivial subbundle OX ⊂ Π. The next piece in the filtration, 875-20

π≤1 = C1 ⊕ Cb−1, gives rise to a rank two subbundle of Π, which we denote by B. It dual bundle is an extension

∗ (17) 0 → Ω → B → OX → 0.

∗ Our universal section Y gives rise to an End Π –valued section of B | × , which projects x x Dx onto the section End Πx–valued section of OX equal to Id (this follows from the vacuum axiom). Explicit computation shows that the space of sections of B∗ which project onto the section 1 of OX (equivalently, splittings of (17)) is canonically isomorphic to the space of affine connections (they may also be described as affine structures, see [Gu]). Thus, we conclude that the field b(z) (or, more precisely, the expression ∂z + b(z)) transforms as an × End Πx–valued affine connection on Dx . Similarly, we obtain connections on G–bundles from the subspace Vk(g)≤1 of the Kac-Moody vertex algebra Vk(g). Let V be a conformal vertex algebra with central charge c. It follows from the definition that the vector space C|0i ⊕ Cω is Aut O–stable. Hence it gives rise to a rank two subbundle Tc of V. Its dual bundle is an extension

2 ∗ (18) 0 → Ω → Tc → OX → 0. ∗ The universal section Yx gives rise to an End Vx–valued section of Tc over the punctured disc, which projects onto Id ∈ End Vx ⊗ Ox. Explicit calculation shows that the space of ∗ sections of Tc projecting onto 1 ∈ Γ(OX ) (equivalently, splittings of (18)) is isomorphic to − 1 3 the space of self–adjoint second order differential operators ρ :Ω 2 → Ω 2 with constant c symbol . Locally, such an operator can be written as c ∂2 + q(z). Operators of this 6 6 z form with symbol 1 are known as projective connections (they may also be described as projective structures, see [Gu]). Thus, we see that for c 6= 0 the field T (z) (called the stress 2 6 tensor in conformal field theory), or more precisely, the expression ∂z + c T (z) transforms as an End Vx–valued projective connection on the punctured disc.

4.4. General twisting property. We have seen in §4.2 that the vertex operation Y is in some sense invariant under the Aut O–action (see formula (16)). Therefore Y gives rise to a well-defined operation on the twist of V by any Aut O–torsor. This “twisting property” follows from the fact that Aut O acts on V by “internal symmetries”, that is by exponentiation of Fourier coefficients of the vertex operator Y (ω, z). It turns out that vertex algebras exhibit a similar twisting property with respect to any group G of internal symmetry, i.e., a group (or more generally, an ind-group) obtained by exponentiation of Fourier coefficients of vertex operators. Using associativity, one can obtain an analogue of formula (16) for the transformations of Y under the action of G. This formula means that we get a well-defined operation on the twist of V by any G–torsor. ∨ For example, let V be a vertex algebra with a bg–structure of level k 6= −h , i.e., one equipped with a homomorphism Vk(g) → V , whose image is not contained in C|0i. 875-21

∨ Recall that Vk(g) is a conformal vertex algebra if k 6= −h (see §3.2). Given such a homomorphism Vk(g) → V , the Fourier coefficients of the corresponding fields Y (ω, z) a and Y (J , z) generate an action of the Lie algebra V ir n bg on V . Suppose that the action of its Lie subalgebra Der0 O n g(O) can be exponentiated to an action of the group Aut O n G(O) on V (then we say that this group acts on V by internal symmetries). Let P be a G–bundle on a smooth curve X. Denote by Pb the principal Aut O n G(O)–bundle over X, whose fiber at x consists of pairs (z, s), where z is a formal coordinate at x and P s is a trivialization of P|Dx . Let V be the Pb–twist of V . Then the vertex operation Y on P P ∗ P V gives rise to a canonical section Y of (V ) | × with values in End V . x Dx x

5. CONFORMAL BLOCKS

5.1. The results of the previous section allow us to assign to a conformal vertex algebra a vector bundle with connection on any smooth curve X, and a family of local structures on it. We can now associate to any compact curve X an invariant of the vertex algebra structure.

Definition 5.1.— Let V be a conformal vertex algebra, X a smooth projective curve, and x a point of X. A linear functional ϕ on Vx is called a conformal block if ϕ(Yx · A) ∈ × ∗ ∗ Γ(Dx , V ) can be extended to a regular section of V on X\x for all A ∈ Vx. ∗ The set of conformal blocks is a vector subspace of Vx, denoted by C(X, x, V ).

∗ Let ϕ ∈ C(X, x, V ), and A ∈ Vx. Denote by ϕA the corresponding section of V over

X\x. If W is an Aut O–stable subspace of V , then ϕA can be projected onto a section ∗ of W over X\x. Note that according to the vacuum axiom, ϕ|0i is actually a regular section of V∗ over the whole X, and so is its projection. In particular, taking as W the two-dimensional subspace of V spanned by |0i and ω, we assign to each conformal block a projective connection on X (if c 6= 0). Denote ϕ(·|0i) as h·iϕ. If we choose a formal 2 6 coordinate z at x, then we can write this projective connection as ∂z + c hT (z)iϕ. If c = 0, 2 then we obtain a quadratic differential hT (z)iϕdz .

∨ Remark 5.2.— Let V be a vertex algebra with bg–structure of level k 6= −h , and P a G–bundle on X. A conformal block twisted by P is by definition a linear functional ϕ on P P P ∗ Vx , such that ϕ(Yx · A) (see §4.4) can be extended to a regular section of (V ) on X\x P P for all A ∈ Vx . We denote the corresponding space by C (X, x, V ).

5.2. Examples.

In the case of the Kac-Moody vertex algebra Vk(g) the Definition 5.1 can be simplified. −1 In this case the subspace (g⊗t )vk is Aut O–invariant, and therefore we have a surjection ∗ ∗ ∗ (Vk(g))  g ⊗ Ω. Hence for each ϕ ∈ C(X, x, π),A ∈ Vx, we obtain a regular g –valued × P a one-form on X\x, whose restriction to Dx is ϕ(J(z) · A)dz, where J(z) = a Ja ⊗ J (z). 875-22

P n Now observe that any f = fnz ∈ g ⊗ Kx gives rise to an operator on Vk(g)x, P fe = Resx(f, J(z))dz = fnbn, which does not depend on the choice of the coordinate z. Since by definition ϕ(J(z) · A)dz extends to a regular one-form on X\x, we obtain from the residue theorem that ϕ(fe· A) = 0 for all f ∈ gx,out = g ⊗ C[X\x]. Hence we obtain a map from C(X, x, Vk(g)) to the space of gx,out–invariant functionals on Vk(g)x.

Lemma 5.3.— The space of conformal blocks C(X, x, Vk(g)) is isomorphic to the space of gx,out–invariant functionals on Vk(g)x.

Thus we recover the common definition of conformal blocks as gx,out–invariants. They have been extensively studied recently because of the relation to the moduli spaces of P G–bundles on curves (see §6.4 below). More generally, we have: C (X, x, Vk(g)) = P P HomgP (Vk(g)x , C), where gx,out is the Lie algebra of sections of P × g over X\x. x,out G The situation becomes more subtle in the case of the Virasoro vertex algebra Virc.

The analogue of gx,out is then the Lie algebra Vect(X\x) of vector fields on X\x. By analogy with the Kac-Moody case, we would like to define a homomorphism Vect(X\x) →

End Virc,x sending ξ(z)∂z ∈ Vect(X\x) to Resx ξ(z)T (z)dz. But it is not clear whether this map is independent of the choice of formal coordinate z, unless c = 0 (when the field 2 6 T (z) transforms as a quadratic differential). According to §4.3, if c 6= 0, then ∂z + c T (z) ∗ is naturally a section of a rank two bundle Tc . Hence it can be paired with sections of the rank two bundle Tc ⊗ Ω (we could split this bundle by choosing a projective structure on X, but we prefer not to do that). One can show that the sheaf Tbc = (Tc ⊗ Ω)/dO, which is an extension of Θ by Ω/dO, has a canonical Lie algebra structure (see [Wi, BS]), × such that V irx = Γ(Dx , Tbc) is isomorphic to the Virasoro algebra and acts on Virc,x. The map Γ(X\x, Tbc) → V irx factors through Vect(X\x), and hence the above map

Vect(X\x) → End Virc,x is well-defined. One can then show that the space of conformal blocks C(X, x, Virc) is isomorphic to the space of Vect(X\x)–invariant functionals on

Virc,x.

5.3. General invariance condition.

For general vertex algebras the analogue of the above invariance condition is defined as follows. Let V →∇ V ⊗ Ω, where V ⊗ Ω is placed in degree 0, be the de Rham complex of the flat vector bundle V, considered as a complex of sheaves on the curve X. Here ∇ is the connection operator defined in §4.2 (recall that ∇ = d + L−1 ⊗ dz). For Σ ⊂ X, 0 denote by HdR(Σ, V ⊗ Ω) the degree 0 cohomology of the restriction of this complex to × 0 Σ. If Σ is affine or Dx , then HdR(Σ, V ⊗ Ω) is simply the quotient of Γ(Σ, V ⊗ Ω) by the image of ∇. × × There is a linear map γ : Γ(Dx , V ⊗ Ω) → End Vx sending µ ∈ Γ(Dx , V ⊗ Ω) to a linear operator Oµ = ResxhYx, µi on Vx. Since the residue of a total derivative vanishes, γ 0 × factors through U(Vx) = HdR(Dx , V ⊗ Ω). If we choose a formal coordinate z at x, then 875-23 we can identify U(Vx) with U(V ), the completion of the span of all Fourier coefficients of all vertex operators Y (A, z),A ∈ V . According to formula (9), U(Vx) is a Lie algebra. Note that in the case of Kac-Moody and Virasoro vertex algebras, U(Vx) contains bgx and V irx, respectively, as Lie subalgebras. 0 Given an open Σ ⊂ X, we have a canonical map HdR(Σ, V ⊗ Ω) → U(Vx). Denote its image by UΣ(Vx). It follows from the Beilinson-Drinfeld chiral algebra formalism (see §7) that UΣ(Vx) is a Lie subalgebra of U(Vx). The image of UΣ(Vx) in End Vx is the span of all operators Oµ for µ ∈ Γ(Σ, V ⊗ Ω). The residue theorem implies: Proposition 5.4.— The space of conformal blocks C(X, x, V ) coincides with the space of UX\x(Vx)–invariant functionals on Vx. Therefore the dual space to the space of conformal blocks is the space of coinvariants

H(X, x, V ) = Vx/UX\x(Vx) · Vx. The definition of the spaces C(X, x, V ) and H(X, x, V ) can be generalized to the case of multiple (distinct) points x1, . . . xn on the curve X, at which we “insert” arbitrary conformal V –modules M1,...,Mn.A V –module M is called conformal if the gradation M operator on M coincides, up to a shift, with the Fourier coefficient L0 of the field YM (ω, z). M Assume for simplicity that the eigenvalues of L0 on all modules Mi are integers. Then each Mi is an Aut O–module, and so we can attach to it a vector bundle Mi on X with a flat connection (a general conformal V –module gives rise to a vector bundle with connection on the space of pairs (x, τx), where x ∈ X and τx is a non-zero tangent vector to X at x). n The space of conformal blocks CV (X, (xi), (Mi))i=1 is then by definition the space of linear functionals ϕ on M1,x1 ⊗ · · · ⊗ Mn,xn , such that for any Ai ∈ Mi,xi , i = 1, . . . , n, there exists a section of V∗ over X \{x , . . . , x }, whose restriction to each D× equals 1 n xi

ϕ(A1 ⊗ · · · ⊗ YMi · Ai ⊗ · · · ⊗ An). Informally, one can say that the local sections of V∗ over the discs around the points, obtained by acting with vertex operators at those points, can be “glued together” into a single meromorphic section of V∗. There is a canonical isomorphism

(19) CV (X, (x1, . . . , xn, y), (M1,...,Mn,V )) ' CV (X, (x1, . . . , xn), (M1,...,Mn)), given by ϕ 7→ ϕ|M1⊗···⊗MN ⊗|0i. Remark 5.5.— Let V be a rational vertex algebra (see §3.4). Then we obtain a functor from the category of pointed algebraic curves with insertions of simple V –modules at the n points to the category of vector spaces, which sends (X, (xi), (Mi)) to CV (X, (xi), (Mi))i=1. This is a version of modular functor corresponding to V [Se, Ga, Z2]. It is known that in the case of rational vertex algebras Lk(g) or Lc(p,q) the spaces of conformal blocks are finite-dimensional. Moreover, the functor can be extended to the category of pointed stable curves with insertions. It then satisfies a factorization property, which expresses the space of conformal blocks associated to a singular curve with a double point in terms 875-24 of conformal blocks associated to its normalization. The same is expected to be true for general rational vertex algebras.

5.4. Functional realization.

n The spaces CV (X, (xi), (Mi))i=1 with varying points x1, . . . , xn can be organized into ◦ a vector bundle on Xn := Xn\∆ (where ∆ is the union of all diagonals) with a flat ∗ connection, which is a subbundle of (M1  ...  Mn) . Consider for example the spaces n CV (X, (xi), (V ))i=1. By (19), all of them are canonically isomorphic to each other and to CV (X, x, V ). Hence the corresponding bundles of conformal blocks are canonically trivial- n ized. For ϕ ∈ CV (X, x, V ), let ϕn denote the corresponding element of CV (X, (xi), (V ))i=1. Let Ai(xi) be local sections of V near xi. Evaluating our conformal blocks on them, we obtain what physicists call the chiral correlation functions

(20) ϕn(A1(x1) ⊗ ... ⊗ An(xn)) ∼ hA1(x1) ...An(xn)iϕ.

Moreover, we can explicitly describe these functions when xi’s are very close to each other, in the neighborhood of a point x ∈ X: if we choose a formal coordinate z at x and denote the coordinate of the point xi by zi, then

(21) ϕn(A1(z1) ⊗ ... ⊗ An(zn)) = ϕ(Y (A1, z1) ...Y (An, zn)|0i). ◦ n n We obtain that the restriction of the n–point chiral correlation function to Dx = Dx \∆ := −1 Spec C[[z1, . . . , zn]][(zi −zj) ]i6=j coincides with the n–point function corresponding to the functional ϕ introduced in §2.7. According to Remark 4.2, the matrix elements given by the right hand side of formula ◦ n ∗ n (21) give rise to a horizontal section of (V ) on Dx. Therefore by Proposition 2.7 we ◦ ∗ n n ∗ obtain an embedding Vx ,→ Γ∇(Dx, (V ) ), where Γ∇ stands for the space of horizontal ∗ sections. If ϕ ∈ Vx is a conformal block, then Definition 5.1 implies that the image of ◦ ◦ n n ∗ n ∗ n ϕ ∈ CV (X, x, V ) in Γ∇(Dx, (V ) ) extends to a horizontal section ϕn of (V ) over X . ◦ n n ∗ n ∗ n ∗ Thus, we obtain an embedding CV (X, x, V ) ,→ Γ(X , (V ) ) = Γ∇(X , j∗j (V ) ), ◦ where j : Xn ,→ Xn. Imposing the bootstrap conditions on the diagonals from Proposition 2.7, we obtain a ∗ ∗ n ∗ ∗ (2) quotient Vn of j∗j (V ) (the precise definition of Vn uses the operation Y introduced in Theorem 7.1). Since by construction our sections satisfy the bootstrap conditions, we ∗ n ∗ n ∗ have embeddings Vx ,→ Γ∇(Dx , Vn) and CV (X, x, V ) ,→ Γ∇(X , Vn). A remarkable fact is that these maps are isomorphisms for all n > 1. Thus, using correlation functions we obtain functional realizations of the (dual space of) a vertex algebra and its spaces of conformal blocks.

n ∗ ∗ Remark 5.6.— Note that neither (V ) nor Vn is quasicoherent as an OXn –module. It n is more convenient to work with the dual sheaves V and Vn, which are quasicoherent. 875-25

n The sheaf Vn on X is defined by Beilinson and Drinfeld in their construction of factoriza- ∗ tion algebra corresponding to V (see §7). Note that the space Γ∇(X, Vn) is dual to the top n deRham cohomology HdR(X, Vn ⊗ ΩXn ) (which is therefore isomorphic to H(X, x, V )).

In the case of Heisenberg algebra, the functional realization can be simplified, because we can use the sections of the exterior products of the canonical bundle Ω instead of the ∗ 1 2 bundle Π (here we choose 2 b−1 as the conformal vector, so that b(z)dz transforms as a ∗ one-form). Namely, as in §2.7 we assign to ϕ ∈ Πx the collection of polydifferentials

ϕ(b(z1) . . . b(zn)|0i)dz1 . . . dzn.

n n These are sections of Ω (2∆) over Dx , which are symmetric and satisfy the bootstrap condition (12). Let Ω∞ be the sheaf on X, whose sections over U ⊂ X are collections of sections of Ωn(2∆) over U n, for n ≥ 0, satisfying these conditions (the bootstrap condition makes sense globally because of the identification Ω2(2∆)/Ω2(∆) ' O). Then ∗ we have canonical isomorphisms Πx ' Γ(Dx, Ω∞),C(X, x, π) ' Γ(X, Ω∞). The same construction can be applied to the Kac-Moody vertex algebras, see [BD1].

Remark 5.7.— If we consider conformal blocks with general V –module insertions, then ◦ the horizontal sections of the corresponding flat bundle of conformal blocks over Xn will have non-trivial monodromy around the diagonals. For example, in the case of the 1 Kac-Moody vertex algebra Vk(g) and X = P , these sections are solutions of the Knizhnik- Zamolodchikov equations, and the monodromy matrices are given by the R–matrices of the quantum group Uq(g), see [TK, SV2].

6. SHEAVES OF CONFORMAL BLOCKS ON MODULI SPACES

In the previous section we associated the space of conformal blocks to a vertex algebra V , a smooth curve X and a point x of X. Now we want to understand the behavior of these spaces as we move x along X and vary the complex structure on X. More precisely, we wish to organize the spaces of conformal blocks into a sheaf on the moduli space

Mg,1 of smooth pointed curves of genus g (here and below by moduli space we mean the corresponding moduli stack in the smooth topology; D–modules on algebraic stacks are defined in [BB, BD2]). Actually, for technical reasons we prefer to work with the spaces of coinvariants (also called covacua), which are dual to the spaces of conformal blocks. One can construct in a straightforward way a quasicoherent sheaf on Mg,1, whose fiber at (X, x) is the space of coinvariants attached to (X, x). But in fact this sheaf also carries a structure of (twisted) D–module on Mg,1, i.e., we can canonically identify the (projectivizations of) the spaces of coinvariants attached to infinitesimally nearby points of Mg,1 (actually, this D–module always descends to Mg). The key fact used in the proof of this statement is the “Virasoro uniformization” of Mg,1. Namely, the Lie algebra Der K 875-26 acts transitively on the moduli space of triples (X, x, z), where (X, x) are as above and z is a formal coordinate at x; this action is obtained by “gluing” X from Dx and X\x (see Theorem 6.1). In the case when the vertex algebra V is rational, the corresponding twisted D–module on Mg is believed to be coherent, i.e., isomorphic to the sheaf of sections of a vector bundle of finite rank equipped with a projectively flat connection (this picture was first suggested by Friedan and Shenker [FrS]). This is known to be true in the case of the

Kac-Moody vertex algebra Lk(g) [TUY] (see also [So]) and the minimal models of the

Virasoro algebra [BFM]. Moreover, in those cases the D–module on Mg can be extended to a D–module with regular singularities on the Deligne-Mumford compactification Mg, and the dimensions of the fibers at the boundary are equal to those in Mg. This allows one to compute these dimensions by “Verlinde formula” from the dimensions of the spaces of coinvariants attached to P1 with three points. The latter numbers, called the fusion rules, can in turn be found from the matrix of the modular transformation τ 7→ −1/τ acting on the space of characters (see Theorem 3.2). The same pattern is believed to hold for other rational vertex algebras, but as far as I know this has not been proved in general. The construction of the D–module structure on the sheaf of coinvariants is a special case of the general formalism of localization of modules over Harish-Chandra pairs, due to Beilinson-Bernstein [BB], which we now briefly recall.

6.1. Generalities on localization.

A Harish–Chandra pair is a pair (g,K) where g is a Lie algebra, K is a Lie group, such that k = Lie K is embedded into g, and an action Ad of K on g compatible with the adjoint action of K on k ⊂ g and the action of k on g. A(g,K)–action on a scheme Z is the data of an action of g on Z (that is, a homo- morphism ρ : g → ΘZ ), together with an action of K on Z, such that (1) the differential −1 of the K–action is the restriction of the g–action to k, and (2) ρ(Adk(a)) = kρ(a)k .

A(g,K)–action is called transitive if the map g ⊗ OZ → ΘZ is surjective, and simply transitive if this map is an isomorphism. A(g,K)–structure on a scheme S is a principal K–bundle π : Z → S together with a simply transitive (g,K)–action on Z which extends the fiberwise action of K. An example of (g,K)–structure is the homogeneous space S = G/K, where G is a finite–dimensional Lie group, g = Lie G, K is a Lie subgroup of G, and we take the obvious right action of (g,K) on Xb = G. In the finite–dimensional setting, any space with a (g,K)–structure is locally of this form. However, in the infinite-dimensional setting this is no longer so. An example is the (Der O, Aut O)–structure Z = Xb → X on a smooth curve X, which we used above. This example can be generalized to the case when S is an arbitrary smooth scheme. Define Sb to be the scheme of pairs (x,~tx), where x ∈ S and ~tx 875-27 is a system of formal coordinates at x. Set g = Der C[[z1, . . . , zn]],K = Aut C[[z1, . . . , zn]]. Then Sb is a (g,K)–structure on S, first considered by Gelfand, Kazhdan and Fuchs [GKF]. Let V be a (g,K)–module, i.e., a vector space together with actions of g and K satisfying the obvious compatibility condition. Then we define a flat vector bundle V on any variety S with a (g,K)–structure Z. Namely, as a vector bundle, V = Z×V . Since by assumption K the g–action on Z is simply transitive, it gives rise to a flat connection on the trivial vector bundle Z ×V over Z, which descends to V, because the actions of K and g are compatible. In the special case of the (Der O, Aut O)–structure Xb over a smooth curve X we obtain the flat bundle V on X from §4.2. Now consider the case when the g–action is transitive but not simply transitive (there are stabilizers). Then we can still construct a vector bundle V on S equipped with a K– equivariant action of the Lie algebroid gZ = g⊗OZ , but we cannot obtain an action of ΘZ on V because the map a : gZ → ΘZ is no longer an isomorphism. Nevertheless, ΘZ will act on any sheaf, on which Ker a acts by 0. In particular, the sheaf V ⊗OZ / Ker a·(V ⊗OZ ) '

DZ ⊗Ug V gives rise to a K–equivariant D–module on Z, and hence to a D–module on S, denoted ∆(V ) and called the localization of V on S. More generally, suppose that V is a module over a Lie algebra l, which contains g as a Lie subalgebra and carries a compatible K–action. Suppose also that we are given a

K–equivariant Lie algebra subsheaf el of the constant sheaf of Lie algebras l⊗OZ , which is preserved by the natural action of the Lie algebroid gZ . Then ifel contains Ker a, the sheaf V ⊗ OZ /el · (V ⊗ OZ ) is a K–equivariant D–module on Z, which descends to a D–module ∆(e V ) on S.

Let us describe the fibers of ∆(e V ) (considered as an OS–module). For s ∈ S, let Zs be the fiber of Z at s, and ls = Zs × l, Vs = Zs × V . The fibers of el at the points of Zs ⊂ M K K give rise to a well-defined Lie subalgebra els of ls. Then the fiber of ∆(e V ) at s ∈ S is canonically isomorphic to the space of coinvariants Vs/els · Vs.

6.2. Localization on the moduli space of curves.

We apply the above formalism in the case when S is the moduli space Mg,1 of smooth pointed curves of genus g > 1, and Z is the moduli space Mcg,1 of triples (X, x, z), where (X, x) ∈ Mg,1 and z is a formal coordinate at x. Clearly, Mcg,1 is an Aut O–bundle over

Mg,1.

Theorem 6.1 (cf. [ADKP, BS, Ko, TUY]).— Mcg,1 carries a transitive action of Der K compatible with the Aut O–action along the fibers.

The action of the corresponding ind–group Aut K is defined by the “gluing” construc- tion. If (X, x, z) is an R–point of Mcg,1, and ρ ∈ Aut K(R), we construct a new R–point (Xρ, xρ, zρ) of Mcg,1 by “gluing” the formal neighborhood of x in X with X\x with a “twist” by ρ. 875-28

Now we are in the situation of §6.1, with g = Der K and K = Aut O. Denote by A the Lie algebroid Der K⊗O on Mg,1. By construction, the kernel of the corresponding b Mcg,1 c homomorphism a : A → Θ is the subsheaf Aout of A, whose fiber at (X, x) ∈ Mg,1 is

Vect(X\x). Applying the construction of §6.1 to V = Vir0 (the Virasoro vertex algebra with c = 0), we obtain a D–module ∆(Vir0) on Mg,1, whose fibers are the spaces of coinvariants Vir0 / Vect(X\x) · Vir0. More generally, let V be an arbitrary conformal vertex algebra with central charge 0.

Then it is a module over the Lie algebra l = U(V ) from §5.3. Let el = U(V )out be the subsheaf of U(V )⊗O , whose fiber at (X, x, z) ∈ Mg,1 equals UX\x(V ) (see §5.3). Note b Mcg,1 that U(V ) contains Der K, and U(V )out contains Aout. Moreover, we have:

Lemma 6.2.— U(V )out is preserved by the action of the Lie algebroid A.

This is equivalent to the statement that UX\x(V ) and UXρ\xρ (V ) are conjugate by ρ, which follows from the fact that formula (16) is valid for any ρ ∈ Aut K.

So we are again in the situation of §6.1, and hence we obtain a D–module ∆(e V ) on Mg,1, whose fiber at (X, x) is precisely the space of coinvariants H(X, x, V ) = Vx/UX\x(Vx) · Vx (see §5.3), which is what we wanted. In the case of vertex algebras with non-zero central charge, we need to modify the general construction of §6.1 as follows. Suppose that g has a central extension bg which splits over k, and such that the extension

(22) 0 → OZ → bg ⊗ OZ → gZ → 0

0 splits over the kernel of a : gZ → ΘZ . Then the quotient gZ of bg ⊗ OZ by Ker a is an extension of ΘZ by OZ , with a natural Lie algebroid structure. The enveloping algebra of 0 this Lie algebroid is a sheaf DZ of twisted differential operators (TDO) on Z, see [BB]. 0 0 Further, gZ descends to a Lie algebroid on S, and gives rise to a sheaf of TDO DS. Now if V is a (g,K)–module, we obtain a D0 –module D0 ⊗ V . By construction, it is b Z Z Ubg 0 K–equivariant, and hence descends to a DS–module on S. More generally, suppose that bg is a Lie subalgebra of bl, and we are given a subsheaf el of bl ⊗ OZ , containing Ker a and preserved by the action of bg ⊗ OZ . Then the sheaf V ⊗ OZ /el · V ⊗ OZ is a K–equivariant 0 0 DZ –module on Z, which descends to a DS–module (still denoted by ∆(e V )). Its fibers are the coinvariants Vs/els · Vs. Let V be a conformal vertex algebra with an arbitrary central charge c. Then by the residue theorem the corresponding sequence (22) with bg = V ir splits over Ker a = Aout. Hence there exists a TDO sheaf Dc on Mg,1 and a Dc–module ∆(e V ) on Mg,1 whose fiber at (X, x) is the space of coinvariants H(X, x, V ). Actually, ∆(e V ) descends to Mg. Explicit computation shows that the sheaf of differential operators on the determinant line bundle over Mg corresponding to the sheaf of relative λ–differentials on the universal curve is 2 Dc(λ), where c(λ) = −12λ + 12λ − 2 (see [BS, BFM]). 875-29

It is straightforward to generalize the above construction to the case of multiple points with arbitrary V –module insertions M1,...,Mn. In that case we obtain a twisted D– 0 module on the moduli space Mg,n of n–pointed curves with non-zero tangent vectors at the points.

6.3. Localization on other moduli spaces.

In the previous section we constructed twisted D–modules on the moduli spaces of pointed curves using the interior action of the Harish-Chandra pair (V ir, Aut O) on con- formal vertex algebras and its modules. This construction can be generalized to the case of other moduli spaces if we consider interior actions of other Harish–Chandra pairs.

For example, consider the Kac-Moody vertex algebra Vk(g). It carries an action of the Harish-Chandra pair (bg,G(O)), where G is the simply-connected group with Lie algebra g. The corresponding moduli space is the moduli space MG(X) of G–bundles on a curve

X. For x ∈ X, let McG(X) be the moduli spaces of pairs (P, s), where P is a G–bundle on X, and s is a trivialization of P|Dx . Then McG(X) is a principal G(Ox)–bundle over

MG(X), equipped with a compatible g(Kx)–action. The latter is given by the gluing construction similar to that explained in the proof of Theorem 6.1.

Now we can apply the construction of §6.1 to the (bgx,G(Ox))–module Vk(g)x, and obtain a twisted D–module on MG(X). Its fiber at P ∈ MG(X) is the space of coinvariants P P Vk(g)x/gx,out · Vk(g)x, where gx,out is the Lie algebra of sections of P × g over X\x. G ∨ More generally, let V be a vertex algebra equipped with a bg–structure of level k 6= −h (see §4.4). Then V carries an action of the Harish-Chandra pair (bgx,G(Ox)). In the same way as in the case of the Virasoro algebra, we attach to V a twisted D–module on P MG(X), whose fiber at P is the twisted space of coinvariants H (X, x, V ), dual to the space CP(X, x, V ) defined in Remark 5.2. For the Heisenberg vertex algebra π, the corresponding moduli space is the Jacobian J(X) of X. The bundle Jb(X) over J(X), which parameterizes line bundles together with trivializations on Dx, carries a transitive action of the abelian Lie algebra K. Applying the above construction, we assign to any conformal vertex algebra equipped with an embed- ding π → V a twisted D–module on J(X), whose fiber at L ∈ J(X) is the twisted space of coinvariants HL(X, x, V ). The corresponding TDO sheaf is the sheaf of differential operators acting on the theta line bundle on J(X).

By considering the action of the semi-direct product V irnbg on V we obtain a twisted D- module on the moduli space of curves and G–bundles on them. We can further generalize the construction by inserting modules at points of the curve, etc. 875-30

6.4. Local and global structure of moduli spaces.

In the simplest cases the twisted D–modules obtained by the above localization con- struction are the twisted sheaves D themselves (considered as left modules over them- selves). For example, the sheaf ∆(Vk(g)) on MG(X) is the sheaf Dk of differential opera- tors acting on the kth power of the determinant line bundle LG on MG(X) (the ample gen- erator of the Picard group of MG(X)). The dual space to the stalk of Dk at P ∈ MG(X) ⊗k is canonically identified with the space of sections of LG on the formal neighborhood of P in MG(X). But according to our construction, this space is also isomorphic to the P space of conformal blocks C (X, x, Vk(g)). In particular, the coordinate ring of the formal P deformation space of a G–bundle P on a curve X is isomorphic to C (X, x, V0(g)). Thus, using the description of conformal blocks in terms of correlation functions (see §5.4), we obtain a realization of the formal deformation space and a line bundle on it in terms of polydifferentials on powers of the curve X satisfying bootstrap conditions on the diago- nals (see [BG, BD2, Gi]). On the other hand, if we replace X by Dx, the corresponding ∗ space of polydifferentials becomes Vk(g) (more precisely, its twist by the torsor of formal coordinates at x). Therefore the vertex algebra Vk(g) may be viewed as the local object responsible for deformations of G–bundles on curves.

Similarly, the Virasoro vertex algebra Virc and the Heisenberg vertex algebra π may be viewed as the local objects responsible for deformations of curves and line bundles on curves, respectively. Optimistically, one may hope that any one–parameter family of “Verma module type” vertex algebras (such as Virc or Vk(g)) is related to a moduli space of curves with some additional structures. It would be interesting to identify the deformation problems related to the most interesting examples of such vertex algebras (for instance, the W–algebras of §3.7), and to construct the corresponding global moduli spaces. One can attach a “formal moduli space” to any vertex algebra V by taking the double quotient of the ind–group, whose Lie algebra is U(V ), by its “in” and “out” subgroups. But this moduli space is very big, even in the familiar examples of Virasoro and Kac-Moody algebras. The question is to construct a smaller formal moduli space with a line bundle, whose space of sections is isomorphic to the space of conformal blocks C(X, x, V ). While “Verma module type” vertex algebras are related to the local structure of moduli spaces, rational vertex algebras (see §3.4) can be used to describe the global structure.

For instance, the space of conformal blocks C(X, x, Lk(g)) ' Homgout (Lk(g), C), is ⊗k isomorphic to Γk = Γ(MG(X), LG ), see [BL, Fa, KNR]. The space C(X, x, Lk(g)) satisfies a factorization property: its dimension does not change under degenerations of the curve into curves with nodal singularities. This property allows one to find this dimension by 1 computing the spaces of conformal blocks in the case of P and three points with Lk(g)– module insertions (fusion rules). The resulting formula for dim C(X, x, Lk) = dim Γk is called the Verlinde formula, see [TUY, So]. 875-31

Since LG is ample, we can recover the moduli space of semi-stable G–bundles on X as the Proj of the graded ring ⊕k≥0ΓNk = ⊕k≥0C(X, x, LNk(g)) for large N (the ring of “non-abelian theta functions”). Thus, if we could define the product on conformal blocks in a natural way, we would obtain a description of the moduli space. Using the correlation function description of conformal blocks Feigin and Stoyanovsky [FS] have identified the space C(X, x, Lk(g)) with the space of sections of a line bundle on the power of X satisfying certain conditions. For example, when g = sl2 it is the space of sections of the line bundle Ω(2nx)nk over Xnk, which are symmetric and vanish on the diagonals of codimension k (here x ∈ X, and n ≥ g). In these terms the multiplication

Γk × Γl → Γk+l is just the composition of exterior tensor product and symmetrization of these sections. Thus we obtain an explicit description of the coordinate ring of the moduli space of semi-stable SL2–bundles. On the other hand, replacing in the above description ∗ X by Dx, we obtain a functional realization of Lk(sl2) . Similarly, the space of conformal blocks corresponding to the lattice vertex superalgebra V√ (see §3.3) may be identified with the space of theta functions of order N on J(X), NZ so we obtain the standard “functional realization” of the Jacobian J(X).

6.5. Critical level.

∨ When k = −h (which is called the critical level), the vertex algebra Vk(g) is not conformal. Nevertheless, V−h∨ (g) still carries an action of Der O which satisfies the same properties as for non-critical levels. Therefore we can still attach to V−h∨ a twisted D– 0 module on MG(X) (but we cannot vary the curve). This D–module is just the sheaf D of differential operators acting on the square root of the canonical bundle on MG(X) (see [BD2]). Recall from §3.6 that the center of a vertex algebra V is the coset vertex algebra of the pair (V,V ). This is a commutative vertex subalgebra of V . The center of Vk(g) is simply its subspace of g[[z]]–invariants. It is easy to show that this subspace equals Cvk ∨ if k 6= −h . In order to describe the center z(bg) of V−h∨ (g) we need the concept of opers, due to Beilinson and Drinfeld [BD2].

Let Gad be the adjoint group of g, and Bad be its Borel subgroup. By definition, a g–oper on a smooth curve X is a Gad–bundle on X with a (flat) connection ∇ and a reduction to Bad, satisfying a certain transversality condition [BD2]. For example, an sl2–oper is the same as a projective connection. The set of g–opers on X is the set of ` di+1 points of an affine space, denoted Opg(X), which is a torsor over ⊕i=1Γ(X, Ω ), where di’s are the exponents of g. Denote by Ag(X) the ring of functions on Opg(X). The above definition can also be applied when X is replaced by the disc D = Spec C[[z]] (see [DS]). The corresponding ring Ag(D) has a natural (Der O, Aut O)–action, and is therefore a commutative vertex algebra (see §2.2). It is isomorphic to a limit of the W– vertex algebra Wk(g) as k → ∞ (see §3.7), and because of that it is called the classical 875-32

W–algebra corresponding to g. For example, Asl2 (D) is a limit of Virc as c → ∞. The following result was conjectured by Drinfeld and proved in [FF3].

Theorem 6.3.— The center z(bg) of V−h∨ (g) is isomorphic, as a commutative vertex L algebra with Der O–action, to ALg(D), where g is the Langlands dual Lie algebra to g.

In fact, more is true: both of these commutative vertex algebras carry what can be called Poisson vertex algebra (or coisson algebra, in the terminology of [BD3]) structures, which are preserved by this isomorphism. The localization of z(g) on M (X) is the O–module H(X, x, z(g))⊗O (since g does b G b MG(X) b not preserve z(bg) ⊂ V−h∨ (g), this sheaf does not carry a D–module structure). By func- toriality, the embedding z(bg) → V−h∨ (g) of vertex algebras gives rise to a homomorphism of O–modules H(X, x, z(g)) ⊗ O → D0. Theorem 6.3 implies that H(X, x, z(g)) ' b MG(X) b H(X, x, ALg(D)). As shown in [BD3], the space of coinvariants of a commutative vertex algebra A is canonically isomorphic to the ring of functions on Γ∇(X, Spec A). In our case we obtain: Γ∇(X, Spec ALg) ' OpLg(X), and so H(X, x, ALg(D)) ' ALg(X). Passing to global sections we obtain:

0 Corollary 6.4.— There is a homomorphism of algebras ALg(X) → Γ(MG(X), D ).

Beilinson and Drinfeld have shown in [BD2] that this map is an embedding, and if G is simply-connected, it is an isomorphism. In that case, each LG–oper ρ gives us a point 0 of the spectrum of the commutative algebra Γ(MG(X), D ), and hence a homomorphism ρ : Γ(M (X), D0) → . The D0–module D0/(D0 · m ) on M (X), where m is the kernel e G C ρe G ρe of ρe, is holonomic. It is shown in [BD2] that the corresponding untwisted D–module is a Hecke eigensheaf attached to ρ by the geometric Langlands correspondence.

6.6. Chiral deRham complex.

Another application of the general localization pattern of §6.1 is the recent construction, due to Malikov, Schechtman and Vaintrob [MSV], of a sheaf of vertex superalgebras on any smooth scheme S called the chiral deRham complex. Under certain conditions, this sheaf has a purely even counterpart, the sheaf of chiral differential operators. Here we give a brief review of the construction of these sheaves. N Let ΓN be the Weyl algebra associated to the symplectic vector space C((t)) ⊕ N ∗ (C((t))dt) . It has topological generators ai,n, ai,n, i = 1,...,N; n ∈ Z, with relations ∗ ∗ ∗ [ai,n, aj,m] = δi,jδn,−m, [ai,n, aj,m] = [ai,n, aj,m] = 0.

Denote by HN the Fock representation of ΓN generated by the vector |0i, satisfying ∗ ai,n|0i = 0, n ≥ 0; ai,m|0i = 0, m > 0. This is a vertex algebra, such that

X −n−1 ∗ ∗ X ∗ −n Y (ai,−1|0i, z) = a(z) = ai,nz ,Y (ai,0|0i, z) = a (z) = ai,nz . n∈Z n∈Z 875-33

V• Let N be the fermionic analogue of HN defined in the same way as in §3.7. This is ∗ the Fock module over the Clifford algebra with generators ψi,n, ψi,n, i = 1, . . . , N, n ∈ Z, equipped with an additional Z–gradation. It is shown in [MSV] that the vertex algebra ∗ • V• structure on HN may be extended to HbN = HN ⊗ [[a ]]. Let Ωb = HbN ⊗ . It ∗ C i,0 N N C[ai,0] P ∗ carries a differential d = Res i ai(z)ψi (z), which makes it into a complex. 1 Denote by WN (resp., ON ,ΩN ) the topological Lie algebra of vector fields (resp., ring • of functions, module of differentials) on Spec C[[t1, . . . , tN ]]. Define a map WN → U(ΩbN ) • (the completion of the Lie algebra of Fourier coefficients of fields from ΩbN ) sending

N ! ∗ X ∗ ∗ f(ti)∂tj 7→ Res : f(ai (z))aj(z) : + :(∂tk f)(ai (z))ψk(z)ψj(z): . k=1 Explicit computation shows that it is a homomorphism of Lie algebras (this is an example of “supersymmetric cancellation of anomalies”) [FF2, MSV]. We obtain a WN –action on • ΩbN , which can be exponentiated to an action of the Harish-Chandra pair (WN , Aut ON ). Furthermore, because the (WN , Aut ON )–action comes from the residues of vertex opera- tors, it preserves the vertex algebra structure on V (so the group Aut ON is an example of the group of internal symmetries from §4.4). Recall from §6.1 that any smooth scheme

S of dimension N carries a canonical (WN , Aut ON )–structure Sb. Applying the construc- • tion of §6.1, we attach to ΩbN a D–module on S. The sheaf of horizontal sections of this D–module is a sheaf of vertex superalgebras (with a structure of complex) on S. This is the chiral deRham complex of S, introduced in [MSV]. For its applications to mirror symmetry and elliptic cohomology, see [BoL, Bo1].

Now consider a purely bosonic analogue of this construction (i.e., replace ΩbN with HbN ). The Lie algebra U(HbN ) of Fourier coefficients of the fields from HbN has a filtration ≤i ≤1 0 AN , i ≥ 0, by the order of ai,n’s. Note that AN is a Lie subalgebra of U(HbN ) and AN ≤1 0 is its (abelian) ideal. Denote by TN the Lie algebra AN /AN . Intuitively, U(HbN ) is the algebra of differential operators on the topological vector space C((t))N with the standard (0) filtration, A (resp., TN ) is the space of functions (resp., the Lie algebra of vector fields) N on it, and HbN is the module of “delta–functions supported on C[[t]] ”. In contrast to the finite-dimensional case, the extension

0 ≤1 0 → AN → AN → TN → 0

does not split [FF2]. Because of that, the naive map WN → U(HbN ) sending f(ti)∂tj to ∗ Res : f(ai (z))aj(z) : is not a Lie algebra homomorphism. Instead, we obtain an extension

1 (23) 0 → ΩN /dON → WfN → WN → 0

1 ≤1 of WN by its module ΩN /dON , which is embedded into AN by the formula g(ti)dtj 7→ ∗ ∗ Res g(ai (z))∂zaj (z). If the sequence (23) were split, we would obtain a (WN , Aut ON )– action on HbN and associate to HbN a D–module on S in the same way as above. 875-34

1 But in reality the extension (23) does not split for N > 1 (note that Ω1/dO1 = 0). Hence in general we only have an action on HbN of the Harish-Chandra pair (WfN , AutgON ), 1 where AutgON is an extension of Aut ON by the additive group ΩN /dON . This action again preserves the vertex algebra structure on V . To apply the construction of §6.1, we need a lifting of the (WN , Aut ON )–structure Sb on S to a (WfN , AutgON )–structure. Such liftings 1 2 form a gerbe on S (in the analytic topology) with the lien ΩS/dOS ' ΩS,cl, the sheaf of 2 2 closed holomorphic two-forms on S. The equivalence class of this gerbe in H (S, ΩS,cl) is computed in [GMS]. If this class equals 0, then we can lift Sb to an (WfN , AutgON )– 1 2 structure Se on S, and the set of isomorphism classes of such liftings is an H (S, ΩS,cl)– torsor. Applying the construction of §6.1, we can then attach to HbN a D–module on S. The sheaf of horizontal sections of this D–module is a sheaf of vertex algebras on S. This is the sheaf of chiral differential operators on S (corresponding to Se) introduced in [MSV, GMS]. As shown in [GMS], the construction becomes more subtle in Zariski 2 3 topology; in this case one naturally obtains a gerbe with the lien ΩS → ΩS,cl. In the case when S is the flag manifold G/B of a simple Lie algebra g, one can construct the sheaf of chiral differential operators in a more direct way. The action of g on the flag manifold G/B gives rise to a homomorphism g((t)) → TN , where N = dim G/B. It is ≤1 ∨ shown in [Wa, FF1, FF2] that it can be lifted to a homomorphism bg → AN of level −h (Wakimoto realization). Moreover, it gives rise to a homomorphism of vertex algebras V−h∨ (g) → HN . Hence we obtain an embedding of the constant subalgebra g of bg into WfN , and a (g,B)–action on HbN . On the other hand, G/B has a natural (g,B)–structure, namely G. Applying the localization construction to the Harish-Chandra pair (g,B) instead of (WfN , AutgON ), we obtain the sheaf of chiral differential operators on G/B – note that it is uniquely defined in this case (see [MSV]).

7. CHIRAL ALGEBRAS

In §4.2, we gave a coordinate independent description of the vertex operation Y (·, z). The formalism of chiral algebras invented by Beilinson and Drinfeld [BD3] (see [G] for a review) is based on a coordinate–free realization of the operator product expansion (see

§2.6). In the definition of Yx we acted by Y (A, z) on B, placed at the fixed point x ∈ X. In order to define the OPE invariantly, we need to let x move along X as well. This is one as follows. Recall that to each conformal vertex algebra V we have assigned a vector bundle V on any smooth curve X. Choose a formal coordinate z at x ∈ X, and use 2 it to trivialize V|Dx . Then V  V(∞∆) is a sheaf on Dx = Spec C[[z, w]] associated to −1 2 the C[[z, w]]–module V ⊗ V [[z, w]][(z − w) ]. Let ∆!V be a sheaf on Dx associated to the C[[z, w]]–module V [[z, w]][(z − w)−1]/V [[z, w]]. Recall from §4.2 that we have a flat connection on V. Hence V is a vector bundle with a flat connection, i.e., a D–module, on Dx. The sheaves V  V(∞∆) and ∆!V are not vector bundles, but they have natural 875-35

2 structures of D–modules on Dx (independent of the choice of z). Because of that, we switch to the language of D–modules. The following result is proved in the same way as in Theorem 4.1 (see also [HL]).

(2) Theorem 7.1.— Define a map Yx : V  V(∞∆) → ∆!V by the formula (2) Yx (f(z, w)A  B) = f(z, w)Y (A, z − w) · B mod V [[z, w]]. (2) Then Yx is a homomorphism of D–modules, which is independent of the choice of the coordinate z.

At this point it is convenient to pass to the right D–module on X corresponding to the left D–module V. Recall that if F is a left D module on a smooth variety Z, then r F := F ⊗ ΩZ , where ΩZ denotes the canonical sheaf on Z, is a right D–module on Z. Denote ∆ : X,→ X2, j : X2\∆(X) ,→ X2.

Corollary 7.2.— The vertex algebra structure on V gives rise to a homomorphism of 2 ∗ r r r right D–modules on X , µ : j j∗V  V → ∆!V satisfying the following conditions: • (skew-symmetry) µ(f(x, y)A  B) = −σ12 ◦ µ(f(y, x)B  A); • (Jacobi identity) µ(µ(f(x, y, z)A  B)  C) + σ123 ◦ µ(µ(f(y, z, x)B  C)  A) −1 +σ123 ◦ µ(µ(f(z, x, y)C  A)  B) = 0. • (vacuum) we are given an embedding Ω ,→ Vr compatible with the natural homo- ∗ morphism j∗j Ω  Ω → ∆!Ω. Beilinson and Drinfeld define a chiral algebra on a curve X as a right D–module A ∗ on X equipped with a homomorphism j∗j A  A → ∆!A satisfying the conditions of Corollary 7.2 (see [BD3, G]). The axioms of chiral algebra readily imply that for any chiral algebra A, its deRham • cohomology HdR(Σ, A) is a (graded) Lie algebra for any open Σ ⊂ X. In particular, for 0 r an affine open Σ ⊂ X we obtain the result mentioned in §5.3 that HdR(Σ, V ) is a Lie algebra. r 2 Define the right D–module V2 on X as the kernel of the homomorphism µ, and r −1 let V2 = V2 ⊗ ΩX2 be the corresponding left D–module. There is a canonical isomor- 2 2 r phism H(X, x, V ) ' HdR(X , V2) (see [G], Prop. 5.1). It is dual to the isomorphism 2 ∗ C(X, x, V ) ' Γ∇(X , V2) mentioned in §5.4. Beilinson and Drinfeld give a beautiful description of chiral algebras as factorization algebras. By definition, a factorization algebra is a collection of quasicoherent O–modules n Fn on X , n ≥ 1, satisfying a factorization condition, which essentially means that the

fiber of Fn at (x1, . . . , xn) is isomorphic to ⊗s∈S(F1)s, where S = {x1, . . . , xn}. For ∗ ∗ ∗ example, j F2 = j (F1  F1) and ∆ F2 = F1. Intuitively, such a collection may be viewed as an O–module on the Ran space R(X) of all finite non-empty subsets of X. In addition, it is required that F1 has a global section (unit) satisfying natural properties. It is proved 875-36 in [BD3] that under these conditions each Fn is automatically a left D–module. Moreover, r the right DX –module F1 = F1 ⊗ ΩX acquires a canonical structure of chiral algebra, with ∗ r r ∗ r ! r r µ given by the composition j j∗F1  F1 = j j∗F2 → ∆!∆ F2 = ∆!F . In fact, there is an equivalence between the categories of factorization algebras and chiral algebras on X [BD2]. r n For a chiral algebra V , the corresponding sheaf Fn on X is the sheaf Vn mentioned in §5.4 (its dual is the sheaf of chiral correlation functions). Using these sheaves, Beilinson ch r r and Drinfeld define “chiral homology” Hi (X, V ), i ≥ 0, of a chiral algebra V . Intu- itively, this is (up to a change of cohomological dimension) the deRham cohomology of the factorization algebra corresponding to V, considered as a D–module on the Ran space r n n R(X). The 0th chiral homology of V is isomorphic to HdR(X , Vn ⊗ ΩXn ) for all n > 1 and hence to the space of coinvariants H(X, x, V ). The full chiral homology functor may therefore be viewed as a derived coinvariants functor. As an example, we sketch the Beilinson-Drinfeld construction of the factorization al- gebra corresponding to the Kac-Moody vertex algebra V0(g) [BD2] (see also [G]). Let n Grn be the ind-scheme over X whose fiber Grx1,...,xn over (x1, . . . , xn) ∈ X is the moduli space of pairs (P, t), where P is a G–bundle on X, and t is its trivialization on X\{x1, . . . , xn} (e.g., the fiber of Gr1 over any x ∈ X is isomorphic to the affine

Grassmannian G(K)/G(O)). Let e be the section of Grn corresponding to the trivial G– bundle, and Ax1...,xn be the space of delta–functions on Grx1,...,xn supported at e. These n are fibers of a left D–module An on X . Claim: {An} form a factorization algebra. The factorization property for {An} follows from the factorization property of the ind-schemes Q Grn: namely, Grx1,...,xn = s∈S Grs, where S = {x1, . . . , xn}. The chiral algebra corre- r sponding to the factorization algebra {An} is V0(g) . To obtain the factorization algebra r corresponding to Vk(g) with k 6= 0, one needs to make a twist by a line bundle on Gr1. A spectacular application of the above formalism is the Beilinson-Drinfeld construction of the chiral Hecke algebra associated to a simple Lie algebra g and a negative integral ∨ L level k < −h . The corresponding vertex algebra Sk(g) is a module over bg × G: ∗ L Sk(g) = ⊕λ∈LP + Mλ,k ⊗ Vλ , where Vλ is a finite-dimensional G–module with highest weight λ, and Mλ,k is the irreducible highest weight bg–module of level k with highest weight (k + h∨)λ. Here LG is the group of adjoint type corresponding to Lg, and LP + is the set of its dominant weights; we identify the dual h∗ of the Cartan subalgebra of g L ∗ with h = h using the normalized invariant inner product on g. In particular, Sk(g) is a module over Vk(g) = M0,k. Note that using results of [KL] one can identify the tensor L category of G–modules with a subcategory of the category O of bg–modules of level k, so that Vλ corresponds to Mλ,k. The analogues of Sk(g) when g is replaced by an abelian

Lie algebra with a non-degenerate inner product are lattice vertex algebras VL from §3.4. r As a chiral algebra, the chiral Hecke algebra is the right D–module Sk(g) on X corresponding to the twist of Sk(g) by Xb. The corresponding factorization algebra is 875-37 constructed analogously to the above construction of Vk(g) using certain irreducible D– modules on Gr1 corresponding to Vλ; see [G] for the construction in the abelian case (it is also possible to construct a vertex algebra structure on Sk(g) directly, using results of L r [KL]). Moreover, for any flat G–bundle E on X, the twist of Sk(g) by E with respect L to the G–action on Sk(g) is also a chiral algebra on X. Conjecturally, the complex of sheaves on MG(X) obtained by localization of this chiral algebra (whose fibers are its chi- ral homologies) is closely related to the automorphic D–module that should be attached to E by the geometric Langlands correspondence.

REFERENCES

[ADKP] E. ARBARELLO, C. DECONCINI, V. KAC and C. PROCESI – Moduli spaces of curves and representation theory, Comm. Math. Phys. 117 (1988) 1–36. [BL] A. BEAUVILLE and Y. LASZLO – Conformal blocks and generalized theta func- tions, Comm. Math. Phys. 164 (1994) 385–419. [BB] A. BEILINSON and J. BERNSTEIN – A Proof of Jantzen Conjectures, Advances in Soviet Mathematics 16, Part 1, pp. 1–50, AMS 1993. [BD1] A. BEILINSON and V. DRINFELD – Affine Kac–Moody algebras and polydiffer- entials, Int. Math. Res. Notices 1 (1994) 1–11. [BD2] A. BEILINSON and V. DRINFELD – Quantization of Hitchin’s Integrable System and Hecke eigensheaves. Preprint. [BD3] A. BEILINSON and V. DRINFELD – Chiral Algebras. Preprint. [BFM] A. BEILINSON, B. FEIGIN and B. MAZUR – Introduction to Algebraic Field Theory on Curves. Preprint. [BG] A. BEILINSON and V. GINZBURG – Infinitesimal structure of moduli spaces of G–bundles, Duke Math. J. IMRN 4 (1992) 63–74. [BS] A. BEILINSON and V. SCHECHTMAN – Determinant bundles and Virasoro alge- bras, Comm. Math. Phys. 118 (1988) 651–701. [BPZ] A. BELAVIN, A. POLYAKOV and A. ZAMOLODCHIKOV – Infinite conformal symmetries in two–dimensional quantum field theory, Nucl. Phys. B241 (1984) 333– 380. [B1] R. BORCHERDS – Vertex algebras, Kac–Moody algebras and the monster, Proc. Nat. Acad. Sci. USA 83 (1986) 3068–3071. [B2] R. BORCHERDS – Monstrous moonshine and monstrous Lie superalgebras, Invent. Math. 109 (1992) 405–444. [B3] R. BORCHERDS – Quantum vertex algebras, Preprint math.QA/9903038. [Bo1] L. BORISOV – Introduction to the vertex algebra approach to mirror symmetry, Preprint math.AG/9912195. 875-38

[BoL] L. BORISOV, A. LIBGOBER – Elliptic genera of toric varieties and applications to mirror symmetry, Invent. Math. 140 (2000) 453–485. [dBT] J. de BOER and T. TJIN – The relation between quantum W–algebras and Lie algebras, Comm. Math. Phys. 160 (1994) 317-332. [dFMS] P. di FRANCESCO, P. MATHIEU and D. SENECHAL – Conformal Field The- ory. Springer–Verlag 1997. [D1] C. DONG – Vertex algebras associated with even lattices, J. Algebra 161 (1993) 245–265. [D2] C. DONG – Representations of the moonshine module vertex operator algebra, Con- temp. Math. 175 (1994) 27–36. [DL] C. DONG and J. LEPOWSKY – Generalized vertex algebras and relative vertex operators. Progress in Mathematics 112, Birkh¨auser,1993. [DLM] C. DONG, H. LI and G. MASON – Twisted representations of vertex operator algebras, Math. Ann. 310 (1998) 571–600. [DS] V. DRINFELD and V. SOKOLOV – Lie algebras and KdV type equations, J. Sov. Math. 30 (1985) 1975-2036. [EK] P. ETINGOF and D. KAZHDAN – Quantization of Lie bialgebras. V, Preprint math.QA/9808121. [Fa] G. FALTINGS – A proof of the Verlinde formula, J. Alg. Geom. 3 (1994) 347–374. [FL] V. FATEEV and S. LUKYANOV – The models of two-dimensional conformal quan-

tum field theory with Zn symmetry, Int. J. Mod. Phys. A3 (1988), 507–520. [Fe1] B. FEIGIN – The semi-infinite cohomology of Kac–Moody and Virasoro Lie alge- bras, Russ. Math. Surv. 39, No. 2 (1984) 155–156. [FF1] B. FEIGIN and E. FRENKEL – A family of representations of affine Lie algebras, Russ. Math. Surv. 43, No. 5 (1988) 221–222. [FF2] B. FEIGIN and E. FRENKEL – Affine Kac–Moody algebras and semi–infinite flag manifolds, Comm. Math. Phys. 128 (1990) 161–189. [FF3] B. FEIGIN and E. FRENKEL – Affine Kac–Moody algebras at the critical level and Gelfand–Dikii algebras, Int. Jour. Mod. Phys. A7, Suppl. 1A (1992) 197–215. [FF4] B. FEIGIN and E. FRENKEL – Integrals of Motion and Quantum Groups, in Lect. Notes in Math. 1620, pp. 349–418, Springer–Verlag 1996. [FS] B. FEIGIN and A. STOYANOVSKY – Realization of a modular functor in the space of differentials, and geometric approximation of the moduli space of G–bundles, Funct. Anal. Appl. 28 (1994) 257–275. [FB] E. FRENKEL and D. BEN-ZVI – Vertex algebras and algebraic curves, Mathemat- ical Surveys and Monographs 88, AMS 2001. [FKW] E. FRENKEL, V. KAC and M. WAKIMOTO – Characters and fusion rules for W–algebras via quantized Drinfeld-Sokolov reduction, Comm. Math. Phys. 147 (1992), 295–328. 875-39

[FKRW] E. FRENKEL, V. KAC, A. RADUL and W. WANG – W1+∞ and WN with central charge N, Comm. Math. Phys. 170 (1995) 337–357. [FR] E. FRENKEL and N. RESHETIKHIN – Towards deformed chiral algebras, Preprint q-alg/9706023. [FK] I. FRENKEL and V. KAC – Basic representations of affine Lie algebras and dual resonance models, Invent. Math. 62 (1980) 23–66. [FGZ] I. FRENKEL, H. GARLAND and G. ZUCKERMAN – Semi-infinite cohomology and string theory, Proc. Nat. Acad. Sci. U.S.A. 83 (1986) 8442–8446. [FLM] I. FRENKEL, J. LEPOWSKY and A. MEURMAN – Vertex Operator Algebras and the Monster. Academic Press 1988. [FHL] I. FRENKEL, Y.-Z. HUANG and J. LEPOWSKY – On axiomatic approaches to vertex operator algebras and modules. Mem. Amer. Math. Soc. 104 (1993), no. 494. [FZ] I. FRENKEL and Y. ZHU – Vertex operator algebras associated to representations of affine and Virasoro algebras, Duke Math. J. 60 (1992) 123–168. [FrS] D. FRIEDAN and S. SHENKER – The analytic geometry of two-dimensional con- formal field theory, Nucl. Phys. B281 (1987) 509–545. [G] D. GAITSGORY – Notes on 2D Conformal Field Theory and String Theory, in Quantum fields and strings: a course for mathematicians, Vol. 2, pp. 1017–1089, AMS 1999. [Ga] K. GAWEDZKI – Conformal field theory, S´em.Bourbaki, Exp. 704, Asterisque 177-278 (1989) 95–126. [GKF] I.M. GELFAND, D.A. KAZHDAN and D.B. FUCHS – The actions of infinite– dimensional Lie algebras, Funct. Anal. Appl. 6 (1972) 9–13. [Gi] V. GINZBURG – Resolution of diagonals and moduli spaces, in The moduli space of curves, Progress in Math. 129, pp. 231–266, Birkh¨auser1995. [Go] P. GODDARD – Meromorphic conformal field theory, in Infinite-dimensional Lie algebras and groups, V. Kac (ed.), pp. 556–587, World Scientific 1989. [GKO] P. GODDARD, A. KENT and D. OLIVE – Unitary representations of the Vira- soro and super-Virasoro algebras, Comm. Math. Phys. 103 (1986) 105–119. [GMS] V. GORBOUNOV, F. MALIKOV and V. SCHECHTMAN – Gerbes of chiral differential operators. I, math.AG/9906116; II, math.AG/0003170. [Gu] R. GUNNING – Lectures on Riemann Surfaces. Princeton University Press 1966. [Hu] Y.-Z. HUANG – Two-dimensional conformal geometry and vertex operator algebras. Progress in Math. 148. Birkh¨auser1997. [HL] Y.-Z. HUANG and J. LEPOWSKY – On the D–module and formal variable ap- proaches to vertex algebras, in Topics in geometry, pp. 175–202, Birkh¨auser1996. [K1] V. KAC – Infinite-dimensional Lie algebras, Third Edition. Cambridge University Press 1990. [K2] V. KAC – Vertex Algebras for Beginners, Second Edition. AMS 1998. 875-40

[K3] V. KAC – Formal distribution algebras and conformal algebras, in Proc. XXIIth ICMP, Brisbane, 1994, pp. 80–96, International Press 1999. [KL] D. KAZHDAN and G. LUSZTIG – Tensor structures arising from affine Lie alge- bras IV, J. of AMS 7 (1993) 383–453. [Ko] M. KONTSEVICH – The Virasoro algebra and Teichm¨uller spaces, Funct. Anal. Appl. 21 (1987), no. 2, 156–157. [KNR] S. KUMAR, M.S. NARASIMHAN and A. RAMANATHAN – Infinite Grassman- nians and moduli spaces of G–bundles, Math. Ann. 300 (1993) 395–423. (1) [LW] J. LEPOWSKY and R.L. WILSON – Construction of the affine Lie algebra A1 , Comm. Math. Phys. 62 (1978) 43–53. [Li] H. LI – Local systems of vertex operators, vertex superalgebras and modules, J. Pure Appl. Alg. 109 (1996) 143–195. [LZ] B. LIAN and G. ZUCKERMAN – New perspectives on the BRST-algebraic structure of string theory, Comm. Math. Phys. 154 (1993) 613–646. [MSV] A. MALIKOV, V. SCHECHTMAN and A. VAINTROB – Chiral deRham com- plex, Comm. Math. Phys. 204 (1999) 439–473. [SV2] V. SCHECHTMAN and A. VARCHENKO – Quantum groups and homology of local systems, in Algebraic Geometry and Analytic Geometry, M. Kashiwara and T. Miwa (eds.), pp. 182–191, Springer–Verlag 1991. [Se] G. SEGAL – The Definition of Conformal Field Theory, unpublished manuscript. [So] C. SORGER – La formule de Verlinde, S´em.Bourbaki, Exp. 793, Asterisque 237 (1996) 87–114. [TK] A. TSUCHIYA and Y. KANIE – Vertex operators in conformal field theory on P1 and monodromy represenations of the braid group, in Adv. Stud. Pure Math 16, pp. 297–372, Academic Press 1988. [TUY] A. TSUCHIYA, K. UENO and Y. YAMADA – Conformal field theory on universal family of stable curves with gauge symmetries, Adv. Stud. Pure Math. 19, pp. 459– 566, Academic Press 1989. (1) [Wa] M. WAKIMOTO – Fock representations of affine Lie algebra A1 , Comm. Math. Phys. 104 (1986) 605–609. [Wa] W. WANG – Rationality of Virasoro vertex operator algebras, Duke Math. J. IMRN 7 (1993) 197–211. [Wi] E. WITTEN – Quantum field theory, Grassmannians and algebraic curves, Comm. Math. Phys 113 (1988) 529–600. [Z1] Y. ZHU – Modular invariance of characters of vertex operator algebras, J. AMS 9 (1996) 237–302. 875-41

[Z2] Y. ZHU – Global vertex operators on Riemann surfaces, Comm. Math. Phys. 165 (1994) 485–531.

EDWARD FRENKEL Department of Mathematics University of California Berkeley, CA 94720, USA E-mail : [email protected]